| L(s) = 1 | − 5.49e11·2-s + 2.18e18·3-s + 3.02e23·4-s + 1.09e27·5-s − 1.20e30·6-s − 3.21e33·7-s − 1.66e35·8-s − 4.44e37·9-s − 5.99e38·10-s − 1.71e41·11-s + 6.61e41·12-s − 1.77e44·13-s + 1.76e45·14-s + 2.38e45·15-s + 9.13e46·16-s − 1.36e48·17-s + 2.44e49·18-s + 1.09e50·19-s + 3.29e50·20-s − 7.02e51·21-s + 9.43e52·22-s + 3.43e53·23-s − 3.63e53·24-s − 1.53e55·25-s + 9.78e55·26-s − 2.05e56·27-s − 9.70e56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.311·3-s + 0.5·4-s + 0.268·5-s − 0.220·6-s − 1.33·7-s − 0.353·8-s − 0.902·9-s − 0.189·10-s − 1.25·11-s + 0.155·12-s − 1.77·13-s + 0.943·14-s + 0.0835·15-s + 0.250·16-s − 0.340·17-s + 0.638·18-s + 0.337·19-s + 0.134·20-s − 0.416·21-s + 0.889·22-s + 0.559·23-s − 0.110·24-s − 0.928·25-s + 1.25·26-s − 0.593·27-s − 0.667·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+79/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(0.4099000355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4099000355\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.49e11T \) |
| good | 3 | \( 1 - 2.18e18T + 4.92e37T^{2} \) |
| 5 | \( 1 - 1.09e27T + 1.65e55T^{2} \) |
| 7 | \( 1 + 3.21e33T + 5.79e66T^{2} \) |
| 11 | \( 1 + 1.71e41T + 1.86e82T^{2} \) |
| 13 | \( 1 + 1.77e44T + 1.00e88T^{2} \) |
| 17 | \( 1 + 1.36e48T + 1.60e97T^{2} \) |
| 19 | \( 1 - 1.09e50T + 1.05e101T^{2} \) |
| 23 | \( 1 - 3.43e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 1.06e58T + 3.38e115T^{2} \) |
| 31 | \( 1 + 8.40e57T + 6.57e117T^{2} \) |
| 37 | \( 1 - 3.44e61T + 7.72e123T^{2} \) |
| 41 | \( 1 + 3.58e63T + 2.56e127T^{2} \) |
| 43 | \( 1 + 3.56e64T + 1.10e129T^{2} \) |
| 47 | \( 1 + 5.54e65T + 1.24e132T^{2} \) |
| 53 | \( 1 - 1.57e68T + 1.65e136T^{2} \) |
| 59 | \( 1 - 1.90e69T + 7.89e139T^{2} \) |
| 61 | \( 1 + 3.99e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 1.50e72T + 1.81e144T^{2} \) |
| 71 | \( 1 + 2.14e73T + 1.77e146T^{2} \) |
| 73 | \( 1 - 4.69e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 4.84e74T + 8.17e149T^{2} \) |
| 83 | \( 1 - 1.11e76T + 4.04e151T^{2} \) |
| 89 | \( 1 - 9.33e76T + 1.00e154T^{2} \) |
| 97 | \( 1 + 3.67e78T + 9.01e156T^{2} \) |
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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
−13.49100474710539210018061074641, −12.08961249388756092214693007076, −10.29020991601934041166132855789, −9.426753215084284117513657993375, −7.998876990989004627132088851671, −6.67551828532453018283702236190, −5.23764162671501154573810528053, −3.02578621200864120139659541668, −2.38562019883934547741363505964, −0.32632967595846029099632553899,
0.32632967595846029099632553899, 2.38562019883934547741363505964, 3.02578621200864120139659541668, 5.23764162671501154573810528053, 6.67551828532453018283702236190, 7.998876990989004627132088851671, 9.426753215084284117513657993375, 10.29020991601934041166132855789, 12.08961249388756092214693007076, 13.49100474710539210018061074641
