| L(s) = 1 | + 4·3-s − 4·5-s + 7-s + 10·9-s − 5·11-s − 5·13-s − 16·15-s − 8·17-s + 5·19-s + 4·21-s − 12·23-s + 10·25-s + 20·27-s − 5·29-s − 31-s − 20·33-s − 4·35-s + 14·37-s − 20·39-s − 17·41-s − 9·43-s − 40·45-s − 7·47-s − 19·49-s − 32·51-s − 3·53-s + 20·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 0.377·7-s + 10/3·9-s − 1.50·11-s − 1.38·13-s − 4.13·15-s − 1.94·17-s + 1.14·19-s + 0.872·21-s − 2.50·23-s + 2·25-s + 3.84·27-s − 0.928·29-s − 0.179·31-s − 3.48·33-s − 0.676·35-s + 2.30·37-s − 3.20·39-s − 2.65·41-s − 1.37·43-s − 5.96·45-s − 1.02·47-s − 2.71·49-s − 4.48·51-s − 0.412·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - T + 20 T^{2} - 17 T^{3} + 183 T^{4} - 17 p T^{5} + 20 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 5 T + 26 T^{2} + 95 T^{3} + 411 T^{4} + 95 p T^{5} + 26 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 5 T + 31 T^{2} + 70 T^{3} + 341 T^{4} + 70 p T^{5} + 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 8 T + 58 T^{2} + p^{2} T^{3} + 1469 T^{4} + p^{3} T^{5} + 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 5 T + 51 T^{2} - 224 T^{3} + 1409 T^{4} - 224 p T^{5} + 51 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 5 T + 117 T^{2} + 422 T^{3} + 5095 T^{4} + 422 p T^{5} + 117 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + T + 70 T^{2} - 119 T^{3} + 2217 T^{4} - 119 p T^{5} + 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 14 T + 188 T^{2} - 1492 T^{3} + 10941 T^{4} - 1492 p T^{5} + 188 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 17 T + 238 T^{2} + 2123 T^{3} + 16155 T^{4} + 2123 p T^{5} + 238 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 9 T + 136 T^{2} + 807 T^{3} + 7703 T^{4} + 807 p T^{5} + 136 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 7 T + 61 T^{2} - 378 T^{3} - 2059 T^{4} - 378 p T^{5} + 61 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 3 T + 161 T^{2} + 582 T^{3} + 11405 T^{4} + 582 p T^{5} + 161 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 23 T + 255 T^{2} + 1812 T^{3} + 11745 T^{4} + 1812 p T^{5} + 255 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 27 T + 483 T^{2} + 5690 T^{3} + 52101 T^{4} + 5690 p T^{5} + 483 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 20 T + 310 T^{2} + 3443 T^{3} + 32439 T^{4} + 3443 p T^{5} + 310 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 2 T + 160 T^{2} + 317 T^{3} + 16077 T^{4} + 317 p T^{5} + 160 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 17 T + 394 T^{2} + 4021 T^{3} + 49295 T^{4} + 4021 p T^{5} + 394 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 10 T + 240 T^{2} - 1505 T^{3} + 25403 T^{4} - 1505 p T^{5} + 240 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 18 T + 144 T^{2} - 297 T^{3} - 9529 T^{4} - 297 p T^{5} + 144 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 11 T + 173 T^{2} + 2468 T^{3} + 24581 T^{4} + 2468 p T^{5} + 173 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−6.53131753262318625354383616091, −6.08874624632761653353625403548, −6.05803224664279561536099838101, −5.89452002034208907817143738383, −5.46499317098058167854743023796, −5.12705269036631846528311842602, −5.03439693115723161104719959927, −4.79613870249856371703601441021, −4.78977178237595384017403103864, −4.37532811675645999173018897336, −4.34202849427884746440513559603, −4.20702906396356179995133393907, −4.08217434343408751147064650316, −3.50456003493790340081758617816, −3.34226743677191948371300139983, −3.32128416399293775189043240186, −3.25179373471664519863172722905, −2.73069558246263557573783461711, −2.54571943667250735134139509155, −2.52624886706529091916311138978, −2.45244224726155159119614850906, −1.60060959359977518677525572895, −1.58123603926643528750994292919, −1.55400198425027742295485818317, −1.44622823566160394809383303193, 0, 0, 0, 0,
1.44622823566160394809383303193, 1.55400198425027742295485818317, 1.58123603926643528750994292919, 1.60060959359977518677525572895, 2.45244224726155159119614850906, 2.52624886706529091916311138978, 2.54571943667250735134139509155, 2.73069558246263557573783461711, 3.25179373471664519863172722905, 3.32128416399293775189043240186, 3.34226743677191948371300139983, 3.50456003493790340081758617816, 4.08217434343408751147064650316, 4.20702906396356179995133393907, 4.34202849427884746440513559603, 4.37532811675645999173018897336, 4.78977178237595384017403103864, 4.79613870249856371703601441021, 5.03439693115723161104719959927, 5.12705269036631846528311842602, 5.46499317098058167854743023796, 5.89452002034208907817143738383, 6.05803224664279561536099838101, 6.08874624632761653353625403548, 6.53131753262318625354383616091
