| L(s) = 1 | − 3-s + 4·5-s − 3·9-s + 2·11-s + 9·13-s − 4·15-s − 17-s + 10·25-s + 27-s + 5·29-s + 10·31-s − 2·33-s + 26·37-s − 9·39-s − 8·41-s − 7·43-s − 12·45-s − 16·47-s − 9·49-s + 51-s + 5·53-s + 8·55-s + 11·59-s − 12·61-s + 36·65-s + 14·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 9-s + 0.603·11-s + 2.49·13-s − 1.03·15-s − 0.242·17-s + 2·25-s + 0.192·27-s + 0.928·29-s + 1.79·31-s − 0.348·33-s + 4.27·37-s − 1.44·39-s − 1.24·41-s − 1.06·43-s − 1.78·45-s − 2.33·47-s − 9/7·49-s + 0.140·51-s + 0.686·53-s + 1.07·55-s + 1.43·59-s − 1.53·61-s + 4.46·65-s + 1.66·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.22176885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.22176885\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + T + 4 T^{2} + 2 p T^{3} + 8 T^{4} + 2 p^{2} T^{5} + 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 9 T^{2} + 11 T^{3} + 12 p T^{4} + 11 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 9 T^{2} - 45 T^{3} + 223 T^{4} - 45 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 9 T + 47 T^{2} - 222 T^{3} + 72 p T^{4} - 222 p T^{5} + 47 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + T + 2 p T^{2} + 60 T^{3} + 770 T^{4} + 60 p T^{5} + 2 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 53 T^{2} + 99 T^{3} + 1326 T^{4} + 99 p T^{5} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 5 T + 57 T^{2} - 117 T^{3} + 1351 T^{4} - 117 p T^{5} + 57 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 10 T + 95 T^{2} - 627 T^{3} + 3691 T^{4} - 627 p T^{5} + 95 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 26 T + 365 T^{2} - 3495 T^{3} + 24686 T^{4} - 3495 p T^{5} + 365 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 8 T + 169 T^{2} + 951 T^{3} + 10514 T^{4} + 951 p T^{5} + 169 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 7 T + 140 T^{2} + 726 T^{3} + 8164 T^{4} + 726 p T^{5} + 140 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 16 T + 163 T^{2} + 27 p T^{3} + 9728 T^{4} + 27 p^{2} T^{5} + 163 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 94 T^{2} - 192 T^{3} + 5564 T^{4} - 192 p T^{5} + 94 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 11 T + 181 T^{2} - 1485 T^{3} + 15983 T^{4} - 1485 p T^{5} + 181 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 12 T + 222 T^{2} + 1760 T^{3} + 19671 T^{4} + 1760 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 144 T^{2} - 56 T^{3} + 13614 T^{4} - 56 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 14 T + 283 T^{2} - 2313 T^{3} + 27857 T^{4} - 2313 p T^{5} + 283 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 155 T^{2} - 951 T^{3} + 12530 T^{4} - 951 p T^{5} + 155 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 13 T + 188 T^{2} - 1269 T^{3} + 14206 T^{4} - 1269 p T^{5} + 188 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5 T + 250 T^{2} - 576 T^{3} + 26420 T^{4} - 576 p T^{5} + 250 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 234 T^{2} - 1251 T^{3} + 26986 T^{4} - 1251 p T^{5} + 234 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + T + 266 T^{2} + 240 T^{3} + 36440 T^{4} + 240 p T^{5} + 266 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.86918228468365536484953660132, −5.35663754159239295941950437407, −5.17977972472712252398761188278, −5.01525894624619095952468529945, −4.71931388960751125255176552240, −4.71637496410378927705281529127, −4.63152360298224408268598090043, −4.47821603527042921295288197456, −3.94134610229572868564715399020, −3.74910358382120407459638107822, −3.55108032989879135995386120437, −3.49342428010929785060181115022, −3.39633482732072012794400808890, −2.96602370539487319254466840046, −2.65170040348917930025774665378, −2.61704272673108364192208600936, −2.57944532279571412661545154118, −1.98896125428968593814467709287, −1.85814756511703654651736928136, −1.73029756235817197108940398902, −1.47394163060259197969871854239, −1.11129445785579307463866732960, −0.76185950175554343871135212223, −0.67472958890991076044083240027, −0.53861254490421134619534552084,
0.53861254490421134619534552084, 0.67472958890991076044083240027, 0.76185950175554343871135212223, 1.11129445785579307463866732960, 1.47394163060259197969871854239, 1.73029756235817197108940398902, 1.85814756511703654651736928136, 1.98896125428968593814467709287, 2.57944532279571412661545154118, 2.61704272673108364192208600936, 2.65170040348917930025774665378, 2.96602370539487319254466840046, 3.39633482732072012794400808890, 3.49342428010929785060181115022, 3.55108032989879135995386120437, 3.74910358382120407459638107822, 3.94134610229572868564715399020, 4.47821603527042921295288197456, 4.63152360298224408268598090043, 4.71637496410378927705281529127, 4.71931388960751125255176552240, 5.01525894624619095952468529945, 5.17977972472712252398761188278, 5.35663754159239295941950437407, 5.86918228468365536484953660132
