| L(s) = 1 | + 4·3-s + 2·5-s + 4·7-s + 10·9-s + 6·11-s + 4·13-s + 8·15-s + 2·17-s + 8·19-s + 16·21-s + 6·23-s − 2·25-s + 20·27-s − 4·31-s + 24·33-s + 8·35-s + 4·37-s + 16·39-s + 2·41-s + 8·43-s + 20·45-s + 10·49-s + 8·51-s − 8·53-s + 12·55-s + 32·57-s + 16·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.894·5-s + 1.51·7-s + 10/3·9-s + 1.80·11-s + 1.10·13-s + 2.06·15-s + 0.485·17-s + 1.83·19-s + 3.49·21-s + 1.25·23-s − 2/5·25-s + 3.84·27-s − 0.718·31-s + 4.17·33-s + 1.35·35-s + 0.657·37-s + 2.56·39-s + 0.312·41-s + 1.21·43-s + 2.98·45-s + 10/7·49-s + 1.12·51-s − 1.09·53-s + 1.61·55-s + 4.23·57-s + 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(77.75513438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(77.75513438\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2 T + 6 T^{2} - 6 T^{3} + 2 T^{4} - 6 p T^{5} + 6 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 6 T + 30 T^{2} - 62 T^{3} + 218 T^{4} - 62 p T^{5} + 30 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 4 T + 32 T^{2} - 76 T^{3} + 462 T^{4} - 76 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 70 p T^{5} + 38 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 8 T + 64 T^{2} - 360 T^{3} + 1838 T^{4} - 360 p T^{5} + 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6 T + 74 T^{2} - 334 T^{3} + 2282 T^{4} - 334 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} + 32 T^{3} + 1406 T^{4} + 32 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 244 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 104 T^{2} - 316 T^{3} + 5214 T^{4} - 316 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 214 p T^{5} + 134 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 116 T^{2} - 552 T^{3} + 5526 T^{4} - 552 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 128 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 8 T + 200 T^{2} + 1176 T^{3} + 15710 T^{4} + 1176 p T^{5} + 200 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 136 T^{2} - 544 T^{3} + 8510 T^{4} - 544 p T^{5} + 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 12 T + 244 T^{2} - 1948 T^{3} + 23606 T^{4} - 1948 p T^{5} + 244 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 14 T + 194 T^{2} + 1686 T^{3} + 14330 T^{4} + 1686 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 92 T^{2} + 580 T^{3} + 166 T^{4} + 580 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 340 T^{2} - 3972 T^{3} + 38678 T^{4} - 3972 p T^{5} + 340 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 28 T + 516 T^{2} - 6268 T^{3} + 64454 T^{4} - 6268 p T^{5} + 516 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 494 p T^{5} + 198 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 2572 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.94876842681802943463223805149, −5.36669799792139598047497408324, −5.28670976146792945345026201198, −5.17328305038196755661401865278, −5.11960573391547642931433367934, −4.70328927184368602125225781265, −4.52955898307547444254457731486, −4.30950091538459148338362768842, −4.22869953257173178301919086554, −3.80931027808154322449698375894, −3.65630027891625246858819117150, −3.61085488519252835201475973417, −3.46529821576991210864451654425, −3.26342934315351919819501389243, −2.82444611465341017394632826340, −2.74391406237787420006545598322, −2.58636950919662334244056863167, −2.00991084724042185231184821843, −1.94202874982414095794422111323, −1.91650242368866773830104320676, −1.74984145955902830474913624548, −1.19937897394823555931486436557, −0.951039438768437859479533764236, −0.903944213752841266849082523896, −0.816951820364906522644622961486,
0.816951820364906522644622961486, 0.903944213752841266849082523896, 0.951039438768437859479533764236, 1.19937897394823555931486436557, 1.74984145955902830474913624548, 1.91650242368866773830104320676, 1.94202874982414095794422111323, 2.00991084724042185231184821843, 2.58636950919662334244056863167, 2.74391406237787420006545598322, 2.82444611465341017394632826340, 3.26342934315351919819501389243, 3.46529821576991210864451654425, 3.61085488519252835201475973417, 3.65630027891625246858819117150, 3.80931027808154322449698375894, 4.22869953257173178301919086554, 4.30950091538459148338362768842, 4.52955898307547444254457731486, 4.70328927184368602125225781265, 5.11960573391547642931433367934, 5.17328305038196755661401865278, 5.28670976146792945345026201198, 5.36669799792139598047497408324, 5.94876842681802943463223805149
