| L(s) = 1 | − 8·2-s + 64·4-s + 466·5-s + 328·7-s − 512·8-s − 3.72e3·10-s + 3.43e3·11-s + 2.19e3·13-s − 2.62e3·14-s + 4.09e3·16-s + 3.85e4·17-s − 4.43e4·19-s + 2.98e4·20-s − 2.74e4·22-s + 4.68e4·23-s + 1.39e5·25-s − 1.75e4·26-s + 2.09e4·28-s + 1.47e5·29-s + 3.07e5·31-s − 3.27e4·32-s − 3.08e5·34-s + 1.52e5·35-s − 5.03e5·37-s + 3.54e5·38-s − 2.38e5·40-s − 5.85e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.66·5-s + 0.361·7-s − 0.353·8-s − 1.17·10-s + 0.778·11-s + 0.277·13-s − 0.255·14-s + 1/4·16-s + 1.90·17-s − 1.48·19-s + 0.833·20-s − 0.550·22-s + 0.803·23-s + 1.77·25-s − 0.196·26-s + 0.180·28-s + 1.12·29-s + 1.85·31-s − 0.176·32-s − 1.34·34-s + 0.602·35-s − 1.63·37-s + 1.04·38-s − 0.589·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.707506644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707506644\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 466 T + p^{7} T^{2} \) |
| 7 | \( 1 - 328 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3436 T + p^{7} T^{2} \) |
| 17 | \( 1 - 38542 T + p^{7} T^{2} \) |
| 19 | \( 1 + 44372 T + p^{7} T^{2} \) |
| 23 | \( 1 - 46872 T + p^{7} T^{2} \) |
| 29 | \( 1 - 147690 T + p^{7} T^{2} \) |
| 31 | \( 1 - 307664 T + p^{7} T^{2} \) |
| 37 | \( 1 + 503858 T + p^{7} T^{2} \) |
| 41 | \( 1 + 585050 T + p^{7} T^{2} \) |
| 43 | \( 1 - 55780 T + p^{7} T^{2} \) |
| 47 | \( 1 - 302176 T + p^{7} T^{2} \) |
| 53 | \( 1 + 436478 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1311076 T + p^{7} T^{2} \) |
| 61 | \( 1 + 966986 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3898364 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1153000 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5390086 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4902608 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2222892 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2074346 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9524962 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44855817805680467721097193511, −10.06686933334199265623538682074, −9.015118355124488109856677084480, −8.243076450261356244715740913680, −6.74684345365428660392129412251, −6.07099604093356990180908447106, −4.92288881010334740763279411866, −3.09216006712498197365507623572, −1.80139315698984420812097823819, −1.04300433770319765475791400112,
1.04300433770319765475791400112, 1.80139315698984420812097823819, 3.09216006712498197365507623572, 4.92288881010334740763279411866, 6.07099604093356990180908447106, 6.74684345365428660392129412251, 8.243076450261356244715740913680, 9.015118355124488109856677084480, 10.06686933334199265623538682074, 10.44855817805680467721097193511
