| L(s) = 1 | + 1.56e12·2-s + 3.91e19·3-s + 2.80e22·4-s − 1.92e28·5-s + 6.12e31·6-s − 2.38e34·7-s − 3.73e36·8-s + 1.08e39·9-s − 3.01e40·10-s − 1.58e41·11-s + 1.09e42·12-s − 8.55e44·13-s − 3.73e46·14-s − 7.54e47·15-s − 5.91e48·16-s − 3.20e49·17-s + 1.70e51·18-s − 6.82e51·19-s − 5.40e50·20-s − 9.34e53·21-s − 2.47e53·22-s + 7.90e54·23-s − 1.46e56·24-s − 4.18e55·25-s − 1.33e57·26-s + 2.52e58·27-s − 6.69e56·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 1.85·3-s + 0.0115·4-s − 0.948·5-s + 1.86·6-s − 1.41·7-s − 0.994·8-s + 2.45·9-s − 0.953·10-s − 0.105·11-s + 0.0215·12-s − 0.656·13-s − 1.42·14-s − 1.76·15-s − 1.01·16-s − 0.469·17-s + 2.47·18-s − 1.10·19-s − 0.0109·20-s − 2.63·21-s − 0.106·22-s + 0.559·23-s − 1.84·24-s − 0.101·25-s − 0.660·26-s + 2.70·27-s − 0.0164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(82-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+81/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(41)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.56e12T + 2.41e24T^{2} \) |
| 3 | \( 1 - 3.91e19T + 4.43e38T^{2} \) |
| 5 | \( 1 + 1.92e28T + 4.13e56T^{2} \) |
| 7 | \( 1 + 2.38e34T + 2.83e68T^{2} \) |
| 11 | \( 1 + 1.58e41T + 2.25e84T^{2} \) |
| 13 | \( 1 + 8.55e44T + 1.69e90T^{2} \) |
| 17 | \( 1 + 3.20e49T + 4.63e99T^{2} \) |
| 19 | \( 1 + 6.82e51T + 3.79e103T^{2} \) |
| 23 | \( 1 - 7.90e54T + 1.99e110T^{2} \) |
| 29 | \( 1 + 1.55e58T + 2.84e118T^{2} \) |
| 31 | \( 1 - 1.98e60T + 6.31e120T^{2} \) |
| 37 | \( 1 + 2.31e63T + 1.05e127T^{2} \) |
| 41 | \( 1 - 2.84e65T + 4.32e130T^{2} \) |
| 43 | \( 1 + 1.94e66T + 2.04e132T^{2} \) |
| 47 | \( 1 + 4.81e67T + 2.75e135T^{2} \) |
| 53 | \( 1 - 3.25e69T + 4.63e139T^{2} \) |
| 59 | \( 1 - 5.77e71T + 2.74e143T^{2} \) |
| 61 | \( 1 + 2.57e72T + 4.08e144T^{2} \) |
| 67 | \( 1 - 7.40e73T + 8.16e147T^{2} \) |
| 71 | \( 1 + 1.55e74T + 8.95e149T^{2} \) |
| 73 | \( 1 - 3.73e75T + 8.49e150T^{2} \) |
| 79 | \( 1 + 9.43e76T + 5.10e153T^{2} \) |
| 83 | \( 1 + 7.99e77T + 2.78e155T^{2} \) |
| 89 | \( 1 - 5.72e78T + 7.95e157T^{2} \) |
| 97 | \( 1 + 4.74e79T + 8.48e160T^{2} \) |
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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
−14.82752937933291188581964646227, −13.41833770149742898298836562784, −12.54422035139115870404825776735, −9.653737791483596784349277630697, −8.440244633792372188313802593208, −6.83679832552844065603215939942, −4.34635179080936208021723380729, −3.45206290839293237284940022387, −2.56372543266360161214847492714, 0,
2.56372543266360161214847492714, 3.45206290839293237284940022387, 4.34635179080936208021723380729, 6.83679832552844065603215939942, 8.440244633792372188313802593208, 9.653737791483596784349277630697, 12.54422035139115870404825776735, 13.41833770149742898298836562784, 14.82752937933291188581964646227
