| L(s) = 1 | + 1.07e8·2-s − 2.35e13·3-s − 2.44e16·4-s − 1.10e19·5-s − 2.52e21·6-s − 2.64e23·7-s − 6.50e24·8-s + 3.79e26·9-s − 1.18e27·10-s + 5.45e27·11-s + 5.76e29·12-s − 3.65e30·13-s − 2.84e31·14-s + 2.59e32·15-s + 1.83e32·16-s − 2.98e33·17-s + 4.07e34·18-s + 2.14e34·19-s + 2.70e35·20-s + 6.22e36·21-s + 5.85e35·22-s − 2.26e36·23-s + 1.52e38·24-s − 1.55e38·25-s − 3.92e38·26-s − 4.81e39·27-s + 6.48e39·28-s + ⋯ |
| L(s) = 1 | + 0.566·2-s − 1.78·3-s − 0.679·4-s − 0.662·5-s − 1.00·6-s − 1.52·7-s − 0.950·8-s + 2.17·9-s − 0.375·10-s + 0.125·11-s + 1.21·12-s − 0.849·13-s − 0.862·14-s + 1.18·15-s + 0.141·16-s − 0.433·17-s + 1.23·18-s + 0.146·19-s + 0.450·20-s + 2.71·21-s + 0.0709·22-s − 0.0807·23-s + 1.69·24-s − 0.560·25-s − 0.481·26-s − 2.08·27-s + 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(56-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(28)\) |
\(\approx\) |
\(0.1464030324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1464030324\) |
| \(L(\frac{57}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.07e8T + 3.60e16T^{2} \) |
| 3 | \( 1 + 2.35e13T + 1.74e26T^{2} \) |
| 5 | \( 1 + 1.10e19T + 2.77e38T^{2} \) |
| 7 | \( 1 + 2.64e23T + 3.02e46T^{2} \) |
| 11 | \( 1 - 5.45e27T + 1.89e57T^{2} \) |
| 13 | \( 1 + 3.65e30T + 1.84e61T^{2} \) |
| 17 | \( 1 + 2.98e33T + 4.72e67T^{2} \) |
| 19 | \( 1 - 2.14e34T + 2.14e70T^{2} \) |
| 23 | \( 1 + 2.26e36T + 7.85e74T^{2} \) |
| 29 | \( 1 + 2.17e40T + 2.70e80T^{2} \) |
| 31 | \( 1 - 6.52e40T + 1.05e82T^{2} \) |
| 37 | \( 1 + 1.90e43T + 1.78e86T^{2} \) |
| 41 | \( 1 + 2.69e44T + 5.04e88T^{2} \) |
| 43 | \( 1 + 2.28e44T + 6.93e89T^{2} \) |
| 47 | \( 1 - 1.80e45T + 9.23e91T^{2} \) |
| 53 | \( 1 + 3.76e47T + 6.84e94T^{2} \) |
| 59 | \( 1 - 6.96e48T + 2.49e97T^{2} \) |
| 61 | \( 1 - 2.06e49T + 1.56e98T^{2} \) |
| 67 | \( 1 - 8.05e49T + 2.71e100T^{2} \) |
| 71 | \( 1 - 5.14e50T + 6.59e101T^{2} \) |
| 73 | \( 1 - 2.88e50T + 3.03e102T^{2} \) |
| 79 | \( 1 + 3.83e51T + 2.34e104T^{2} \) |
| 83 | \( 1 + 4.00e52T + 3.54e105T^{2} \) |
| 89 | \( 1 + 1.38e53T + 1.64e107T^{2} \) |
| 97 | \( 1 - 6.05e54T + 1.87e109T^{2} \) |
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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
−18.96290724992049252949636848494, −17.22604379166101424893126275305, −15.72373022710451144447220620411, −12.95118231308152375064742191944, −11.87075427096932695407937365486, −9.846693394851963566904888415547, −6.69359478163537418462844897959, −5.29216002292109809460747630907, −3.81047115268815694039929272110, −0.25509747719462604680015866627,
0.25509747719462604680015866627, 3.81047115268815694039929272110, 5.29216002292109809460747630907, 6.69359478163537418462844897959, 9.846693394851963566904888415547, 11.87075427096932695407937365486, 12.95118231308152375064742191944, 15.72373022710451144447220620411, 17.22604379166101424893126275305, 18.96290724992049252949636848494
