| L(s) = 1 | + 3·2-s + 4·3-s + 7·4-s − 5·5-s + 12·6-s − 2·7-s + 15·8-s + 13·9-s − 15·10-s + 3·11-s + 28·12-s + 13-s − 6·14-s − 20·15-s + 30·16-s − 2·17-s + 39·18-s + 5·19-s − 35·20-s − 8·21-s + 9·22-s − 11·23-s + 60·24-s + 10·25-s + 3·26-s + 30·27-s − 14·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2.30·3-s + 7/2·4-s − 2.23·5-s + 4.89·6-s − 0.755·7-s + 5.30·8-s + 13/3·9-s − 4.74·10-s + 0.904·11-s + 8.08·12-s + 0.277·13-s − 1.60·14-s − 5.16·15-s + 15/2·16-s − 0.485·17-s + 9.19·18-s + 1.14·19-s − 7.82·20-s − 1.74·21-s + 1.91·22-s − 2.29·23-s + 12.2·24-s + 2·25-s + 0.588·26-s + 5.77·27-s − 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.33915688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.33915688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 - 4 T + p T^{2} + 10 T^{3} - 29 T^{4} + 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 51 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 2 T + 7 T^{2} + 70 T^{3} + 441 T^{4} + 70 p T^{5} + 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 5 T + 6 T^{2} + 65 T^{3} - 439 T^{4} + 65 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 11 T + 73 T^{2} + 475 T^{3} + 2796 T^{4} + 475 p T^{5} + 73 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 3 T + 23 T^{2} - 171 T^{3} + 1600 T^{4} - 171 p T^{5} + 23 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 2 T + 27 T^{2} + 160 T^{3} + 1841 T^{4} + 160 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 7 T - 17 T^{2} - 331 T^{3} - 1320 T^{4} - 331 p T^{5} - 17 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 18 T + 137 T^{2} - 990 T^{3} + 7951 T^{4} - 990 p T^{5} + 137 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 11 T + 23 T^{2} - 475 T^{3} - 5824 T^{4} - 475 p T^{5} + 23 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 5 T - 44 T^{2} + 25 T^{3} + 3801 T^{4} + 25 p T^{5} - 44 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 8 T + 123 T^{2} - 766 T^{3} + 10325 T^{4} - 766 p T^{5} + 123 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 17 T + 117 T^{2} + 1165 T^{3} + 13436 T^{4} + 1165 p T^{5} + 117 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 12 T - 17 T^{2} - 666 T^{3} - 3185 T^{4} - 666 p T^{5} - 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 19 T + 78 T^{2} + 265 T^{3} - 2809 T^{4} + 265 p T^{5} + 78 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 11 T^{2} - 510 T^{3} + 4711 T^{4} - 510 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 31 T + 328 T^{2} + 695 T^{3} - 8679 T^{4} + 695 p T^{5} + 328 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 25 T + 286 T^{2} + 3125 T^{3} + 34671 T^{4} + 3125 p T^{5} + 286 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 7 T - 3 T^{2} - 775 T^{3} - 4204 T^{4} - 775 p T^{5} - 3 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−8.327476327954466069240300692905, −7.937778918821113242935221066893, −7.77650341261293666937805513714, −7.75673264749485699244130450210, −7.46219475350064464925271608662, −7.10474165343343630897302819556, −6.91030496582627376149745823970, −6.86480525117073175257257327202, −6.56229426373670731114817845861, −5.98176184169714133112437578946, −5.83485450218696570341102409271, −5.64248929146538169230592537336, −5.02813289489399345307109734550, −4.55132819821740836807914645003, −4.33983440511660870800477205215, −4.25829104550419474155824621447, −4.00995705930072767099514814954, −3.80467397177686493965006306939, −3.41148422120355677972720989768, −3.34182912369276277863363414920, −2.79742739110901506063030177934, −2.70071679982648721834892514909, −2.07387801887005116389827983540, −1.51151222933747429957151518149, −1.38666893043369579620537530184,
1.38666893043369579620537530184, 1.51151222933747429957151518149, 2.07387801887005116389827983540, 2.70071679982648721834892514909, 2.79742739110901506063030177934, 3.34182912369276277863363414920, 3.41148422120355677972720989768, 3.80467397177686493965006306939, 4.00995705930072767099514814954, 4.25829104550419474155824621447, 4.33983440511660870800477205215, 4.55132819821740836807914645003, 5.02813289489399345307109734550, 5.64248929146538169230592537336, 5.83485450218696570341102409271, 5.98176184169714133112437578946, 6.56229426373670731114817845861, 6.86480525117073175257257327202, 6.91030496582627376149745823970, 7.10474165343343630897302819556, 7.46219475350064464925271608662, 7.75673264749485699244130450210, 7.77650341261293666937805513714, 7.937778918821113242935221066893, 8.327476327954466069240300692905
