| L(s) = 1 | − 32·2-s + 640·4-s + 278·5-s + 548·7-s − 1.02e4·8-s − 1.76e3·9-s − 8.89e3·10-s + 7.39e3·11-s − 1.75e4·14-s + 1.43e5·16-s − 2.83e4·17-s + 5.65e4·18-s + 9.98e4·19-s + 1.77e5·20-s − 2.36e5·22-s + 3.33e4·23-s − 7.41e4·25-s + 3.54e4·27-s + 3.50e5·28-s − 9.31e4·29-s + 3.11e5·31-s − 1.83e6·32-s + 9.06e5·34-s + 1.52e5·35-s − 1.13e6·36-s + 9.63e3·37-s − 3.19e6·38-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s + 0.994·5-s + 0.603·7-s − 7.07·8-s − 0.807·9-s − 2.81·10-s + 1.67·11-s − 1.70·14-s + 35/4·16-s − 1.39·17-s + 2.28·18-s + 3.34·19-s + 4.97·20-s − 4.73·22-s + 0.572·23-s − 0.949·25-s + 0.346·27-s + 3.01·28-s − 0.709·29-s + 1.87·31-s − 9.89·32-s + 3.95·34-s + 0.600·35-s − 4.03·36-s + 0.0312·37-s − 9.44·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.978132339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978132339\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 589 p T^{2} - 1312 p^{3} T^{3} + 198200 p^{3} T^{4} - 1312 p^{10} T^{5} + 589 p^{15} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 278 T + 151453 T^{2} - 10092158 p T^{3} + 507231524 p^{2} T^{4} - 10092158 p^{8} T^{5} + 151453 p^{14} T^{6} - 278 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 548 T + 205477 p T^{2} - 1005153868 T^{3} + 1410523587416 T^{4} - 1005153868 p^{7} T^{5} + 205477 p^{15} T^{6} - 548 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 672 p T + 69657359 T^{2} - 362239950472 T^{3} + 1972371186671352 T^{4} - 362239950472 p^{7} T^{5} + 69657359 p^{14} T^{6} - 672 p^{22} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 28316 T + 80288674 p T^{2} + 22995691781952 T^{3} + 723743706605364939 T^{4} + 22995691781952 p^{7} T^{5} + 80288674 p^{15} T^{6} + 28316 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 99888 T + 6302386479 T^{2} - 273388952905776 T^{3} + 9264012062067280120 T^{4} - 273388952905776 p^{7} T^{5} + 6302386479 p^{14} T^{6} - 99888 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 33388 T + 4497107523 T^{2} - 11678709274332 p T^{3} + 15457403175217183288 T^{4} - 11678709274332 p^{8} T^{5} + 4497107523 p^{14} T^{6} - 33388 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 93140 T + 34300522218 T^{2} + 2719238895824544 T^{3} + \)\(84\!\cdots\!75\)\( T^{4} + 2719238895824544 p^{7} T^{5} + 34300522218 p^{14} T^{6} + 93140 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 311160 T + 140644653260 T^{2} - 26779795197492120 T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - 26779795197492120 p^{7} T^{5} + 140644653260 p^{14} T^{6} - 311160 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 9636 T + 9745737498 p T^{2} - 1878062788215984 T^{3} + \)\(50\!\cdots\!59\)\( T^{4} - 1878062788215984 p^{7} T^{5} + 9745737498 p^{15} T^{6} - 9636 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 82892 T + 552926747522 T^{2} + 95958507863966880 T^{3} + \)\(13\!\cdots\!27\)\( T^{4} + 95958507863966880 p^{7} T^{5} + 552926747522 p^{14} T^{6} + 82892 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 569264 T + 675637103839 T^{2} - 373504087629850528 T^{3} + \)\(21\!\cdots\!84\)\( T^{4} - 373504087629850528 p^{7} T^{5} + 675637103839 p^{14} T^{6} - 569264 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 574200 T + 1181775847100 T^{2} + 840326241178871192 T^{3} + \)\(77\!\cdots\!90\)\( T^{4} + 840326241178871192 p^{7} T^{5} + 1181775847100 p^{14} T^{6} + 574200 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1235350 T + 2684236572461 T^{2} - 3308983772395621830 T^{3} + \)\(45\!\cdots\!08\)\( T^{4} - 3308983772395621830 p^{7} T^{5} + 2684236572461 p^{14} T^{6} - 1235350 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 231504 T + 2410408632647 T^{2} + 675371015279844664 T^{3} + \)\(89\!\cdots\!72\)\( T^{4} + 675371015279844664 p^{7} T^{5} + 2410408632647 p^{14} T^{6} + 231504 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 685684 T - 2963552678006 T^{2} + 807290360276097248 T^{3} + \)\(21\!\cdots\!11\)\( T^{4} + 807290360276097248 p^{7} T^{5} - 2963552678006 p^{14} T^{6} + 685684 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 3271056 T + 25389643205903 T^{2} + 59152713001018797288 T^{3} + \)\(23\!\cdots\!16\)\( T^{4} + 59152713001018797288 p^{7} T^{5} + 25389643205903 p^{14} T^{6} + 3271056 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 175012 T + 22437418387579 T^{2} - 7492697774427458180 T^{3} + \)\(26\!\cdots\!04\)\( T^{4} - 7492697774427458180 p^{7} T^{5} + 22437418387579 p^{14} T^{6} - 175012 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7137890 T + 48989573892505 T^{2} - \)\(21\!\cdots\!78\)\( T^{3} + \)\(81\!\cdots\!12\)\( T^{4} - \)\(21\!\cdots\!78\)\( p^{7} T^{5} + 48989573892505 p^{14} T^{6} - 7137890 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 7053952 T + 797675870260 p T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} + \)\(36\!\cdots\!40\)\( p^{7} T^{5} + 797675870260 p^{15} T^{6} + 7053952 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 657288 T + 707565190612 p T^{2} + 83704316584548117240 T^{3} + \)\(16\!\cdots\!26\)\( T^{4} + 83704316584548117240 p^{7} T^{5} + 707565190612 p^{15} T^{6} - 657288 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 11452234 T + 85829145588041 T^{2} + 87845546274720266406 T^{3} - \)\(15\!\cdots\!60\)\( T^{4} + 87845546274720266406 p^{7} T^{5} + 85829145588041 p^{14} T^{6} - 11452234 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 428002 T + 262966365990817 T^{2} - \)\(17\!\cdots\!42\)\( T^{3} + \)\(29\!\cdots\!32\)\( T^{4} - \)\(17\!\cdots\!42\)\( p^{7} T^{5} + 262966365990817 p^{14} T^{6} - 428002 p^{21} T^{7} + p^{28} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.03756303518920195876170155139, −7.01339613694661659856620335418, −6.70821487682950512251273605229, −6.56363104845749508210945193692, −6.46821681891209597350460649470, −5.79855245320963747002061246060, −5.70975673505344896551169212456, −5.57867484772998308280544509253, −5.37460375742095279129330887643, −4.88923701035266010361412830962, −4.44324204928085912764662509741, −4.19496608143432468790937241975, −3.92625149448016333001552643943, −3.20527214781971589569875194324, −3.16836092067020112941704425172, −2.87267500947223221718814603193, −2.74311444586108485849332207696, −2.05549331540689843672386537320, −1.97027879172249492254003579745, −1.60443714002249210641926510294, −1.47826896211398055483076364582, −1.11169766386835853901130513463, −0.77963620483203718354731703557, −0.56658744512281102407946892852, −0.33240100619318930520040079901,
0.33240100619318930520040079901, 0.56658744512281102407946892852, 0.77963620483203718354731703557, 1.11169766386835853901130513463, 1.47826896211398055483076364582, 1.60443714002249210641926510294, 1.97027879172249492254003579745, 2.05549331540689843672386537320, 2.74311444586108485849332207696, 2.87267500947223221718814603193, 3.16836092067020112941704425172, 3.20527214781971589569875194324, 3.92625149448016333001552643943, 4.19496608143432468790937241975, 4.44324204928085912764662509741, 4.88923701035266010361412830962, 5.37460375742095279129330887643, 5.57867484772998308280544509253, 5.70975673505344896551169212456, 5.79855245320963747002061246060, 6.46821681891209597350460649470, 6.56363104845749508210945193692, 6.70821487682950512251273605229, 7.01339613694661659856620335418, 7.03756303518920195876170155139
