| L(s) = 1 | + 3·3-s + 3·5-s − 25·7-s − 29·9-s + 56·11-s + 26·13-s + 9·15-s − 13·17-s + 124·19-s − 75·21-s − 172·23-s − 79·25-s − 120·27-s − 196·29-s − 78·31-s + 168·33-s − 75·35-s − 161·37-s + 78·39-s + 234·41-s − 135·43-s − 87·45-s − 237·47-s − 199·49-s − 39·51-s − 666·53-s + 168·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.268·5-s − 1.34·7-s − 1.07·9-s + 1.53·11-s + 0.554·13-s + 0.154·15-s − 0.185·17-s + 1.49·19-s − 0.779·21-s − 1.55·23-s − 0.631·25-s − 0.855·27-s − 1.25·29-s − 0.451·31-s + 0.886·33-s − 0.362·35-s − 0.715·37-s + 0.320·39-s + 0.891·41-s − 0.478·43-s − 0.288·45-s − 0.735·47-s − 0.580·49-s − 0.107·51-s − 1.72·53-s + 0.411·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 38 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 25 T + 824 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 56 T + 3154 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 13 T + 3280 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 124 T + 16394 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 172 T + 29102 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 196 T + 53710 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 78 T + 40006 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 161 T + 35352 T^{2} + 161 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 234 T + 40498 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 135 T + 12442 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 237 T + 221232 T^{2} + 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 666 T + 406818 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 136 T + 57682 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 146 T + 457466 T^{2} - 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 526778 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 563 T + 787016 T^{2} + 563 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1480 T + 1311326 T^{2} + 1480 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 896 T + 1176270 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1902 T + 2047318 T^{2} + 1902 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 272 T + 341902 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2160 T + 2346718 T^{2} + 2160 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.699161966533209872431477867055, −9.126977453473146178824240407708, −8.749121589623530528510096087557, −8.684541101266551973446063909219, −7.77973703247188839483297146950, −7.61458172548842676941970751040, −7.01686697088383768797679247165, −6.44023316269378989692994759614, −6.06236952559241392372100917906, −5.88332149598400480431352359264, −5.34884814522129355674257243789, −4.54773932915670220074026652530, −3.79467790586929594959321251044, −3.68443769952288369477306483403, −3.04990386458146553001191871842, −2.73563150337890594693401693470, −1.64035502780962391854262197441, −1.48026392733629552604050118012, 0, 0,
1.48026392733629552604050118012, 1.64035502780962391854262197441, 2.73563150337890594693401693470, 3.04990386458146553001191871842, 3.68443769952288369477306483403, 3.79467790586929594959321251044, 4.54773932915670220074026652530, 5.34884814522129355674257243789, 5.88332149598400480431352359264, 6.06236952559241392372100917906, 6.44023316269378989692994759614, 7.01686697088383768797679247165, 7.61458172548842676941970751040, 7.77973703247188839483297146950, 8.684541101266551973446063909219, 8.749121589623530528510096087557, 9.126977453473146178824240407708, 9.699161966533209872431477867055
