| L(s) = 1 | − 1.28e13·2-s − 1.04e21·3-s + 9.24e24·4-s + 2.09e30·5-s + 1.33e34·6-s − 6.15e36·7-s + 1.86e39·8-s + 7.62e41·9-s − 2.67e43·10-s + 2.01e45·11-s − 9.63e45·12-s − 2.57e48·13-s + 7.88e49·14-s − 2.17e51·15-s − 2.52e52·16-s − 3.75e53·17-s − 9.76e54·18-s + 2.54e55·19-s + 1.93e55·20-s + 6.41e57·21-s − 2.57e58·22-s − 1.48e59·23-s − 1.94e60·24-s − 2.08e60·25-s + 3.29e61·26-s − 4.57e62·27-s − 5.69e61·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 1.83·3-s + 0.0597·4-s + 0.822·5-s + 1.88·6-s − 1.06·7-s + 0.967·8-s + 2.35·9-s − 0.846·10-s + 1.00·11-s − 0.109·12-s − 0.899·13-s + 1.09·14-s − 1.50·15-s − 1.05·16-s − 1.12·17-s − 2.42·18-s + 0.602·19-s + 0.0491·20-s + 1.95·21-s − 1.03·22-s − 0.865·23-s − 1.77·24-s − 0.323·25-s + 0.925·26-s − 2.48·27-s − 0.0636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(0.2703448160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2703448160\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.28e13T + 1.54e26T^{2} \) |
| 3 | \( 1 + 1.04e21T + 3.23e41T^{2} \) |
| 5 | \( 1 - 2.09e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 6.15e36T + 3.33e73T^{2} \) |
| 11 | \( 1 - 2.01e45T + 3.99e90T^{2} \) |
| 13 | \( 1 + 2.57e48T + 8.18e96T^{2} \) |
| 17 | \( 1 + 3.75e53T + 1.11e107T^{2} \) |
| 19 | \( 1 - 2.54e55T + 1.78e111T^{2} \) |
| 23 | \( 1 + 1.48e59T + 2.95e118T^{2} \) |
| 29 | \( 1 + 4.28e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 1.20e65T + 5.60e129T^{2} \) |
| 37 | \( 1 + 4.68e67T + 2.71e136T^{2} \) |
| 41 | \( 1 + 3.05e69T + 2.05e140T^{2} \) |
| 43 | \( 1 - 6.50e70T + 1.29e142T^{2} \) |
| 47 | \( 1 + 8.10e72T + 2.96e145T^{2} \) |
| 53 | \( 1 - 7.76e74T + 1.02e150T^{2} \) |
| 59 | \( 1 - 1.60e76T + 1.15e154T^{2} \) |
| 61 | \( 1 + 1.52e76T + 2.10e155T^{2} \) |
| 67 | \( 1 + 2.87e79T + 7.38e158T^{2} \) |
| 71 | \( 1 + 4.02e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 1.81e81T + 1.28e162T^{2} \) |
| 79 | \( 1 + 2.02e82T + 1.24e165T^{2} \) |
| 83 | \( 1 - 3.22e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 5.21e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 8.29e85T + 7.06e172T^{2} \) |
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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
−16.17394453869905487333518481525, −13.30024854977703830208300420798, −11.76016847915129159021609077913, −10.18457878372908688745636276108, −9.481110353908106826807864413115, −6.98586011037652825637697529427, −5.96962465111877705669147893079, −4.46387001692769218690422696867, −1.66669921752842481514526654069, −0.38736706094937514541146427892,
0.38736706094937514541146427892, 1.66669921752842481514526654069, 4.46387001692769218690422696867, 5.96962465111877705669147893079, 6.98586011037652825637697529427, 9.481110353908106826807864413115, 10.18457878372908688745636276108, 11.76016847915129159021609077913, 13.30024854977703830208300420798, 16.17394453869905487333518481525
