L(s) = 1 | − 2.56e15·2-s + 7.24e24·3-s − 3.40e31·4-s − 1.17e36·5-s − 1.85e40·6-s − 1.82e44·7-s + 1.91e47·8-s − 7.27e49·9-s + 3.00e51·10-s + 6.21e54·11-s − 2.46e56·12-s + 9.87e57·13-s + 4.67e59·14-s − 8.48e60·15-s + 8.89e62·16-s + 1.10e64·17-s + 1.86e65·18-s + 1.61e67·19-s + 3.98e67·20-s − 1.32e69·21-s − 1.59e70·22-s − 2.39e71·23-s + 1.38e72·24-s − 2.32e73·25-s − 2.53e73·26-s − 1.43e75·27-s + 6.20e75·28-s + ⋯ |
L(s) = 1 | − 0.402·2-s + 0.647·3-s − 0.838·4-s − 0.235·5-s − 0.260·6-s − 0.783·7-s + 0.739·8-s − 0.581·9-s + 0.0948·10-s + 1.31·11-s − 0.542·12-s + 0.325·13-s + 0.315·14-s − 0.152·15-s + 0.540·16-s + 0.278·17-s + 0.233·18-s + 1.18·19-s + 0.197·20-s − 0.506·21-s − 0.530·22-s − 0.775·23-s + 0.478·24-s − 0.944·25-s − 0.130·26-s − 1.02·27-s + 0.656·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(53)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{107}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 + 2.56e15T + 4.05e31T^{2} \) |
| 3 | \( 1 - 7.24e24T + 1.25e50T^{2} \) |
| 5 | \( 1 + 1.17e36T + 2.46e73T^{2} \) |
| 7 | \( 1 + 1.82e44T + 5.43e88T^{2} \) |
| 11 | \( 1 - 6.21e54T + 2.21e109T^{2} \) |
| 13 | \( 1 - 9.87e57T + 9.20e116T^{2} \) |
| 17 | \( 1 - 1.10e64T + 1.57e129T^{2} \) |
| 19 | \( 1 - 1.61e67T + 1.85e134T^{2} \) |
| 23 | \( 1 + 2.39e71T + 9.58e142T^{2} \) |
| 29 | \( 1 - 4.95e76T + 3.56e153T^{2} \) |
| 31 | \( 1 - 8.41e76T + 3.91e156T^{2} \) |
| 37 | \( 1 - 2.33e82T + 4.58e164T^{2} \) |
| 41 | \( 1 - 3.56e84T + 2.19e169T^{2} \) |
| 43 | \( 1 + 2.88e85T + 3.26e171T^{2} \) |
| 47 | \( 1 - 8.77e86T + 3.71e175T^{2} \) |
| 53 | \( 1 + 1.32e90T + 1.11e181T^{2} \) |
| 59 | \( 1 - 9.30e92T + 8.69e185T^{2} \) |
| 61 | \( 1 + 9.36e93T + 2.88e187T^{2} \) |
| 67 | \( 1 + 4.39e95T + 5.46e191T^{2} \) |
| 71 | \( 1 + 2.93e97T + 2.41e194T^{2} \) |
| 73 | \( 1 + 6.72e97T + 4.45e195T^{2} \) |
| 79 | \( 1 + 7.95e99T + 1.78e199T^{2} \) |
| 83 | \( 1 - 6.25e100T + 3.18e201T^{2} \) |
| 89 | \( 1 - 3.11e102T + 4.85e204T^{2} \) |
| 97 | \( 1 + 3.96e104T + 4.08e208T^{2} \) |
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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
−13.51901272130864475680765935145, −11.82226239066862034035907886284, −9.809514923245685914950115524611, −8.967593162309253346710299641169, −7.76250669663305613210077564119, −5.98083503443993720406932585882, −4.13367633193593308154224536720, −3.13596988314855035476847925613, −1.28103608680477674336644983983, 0,
1.28103608680477674336644983983, 3.13596988314855035476847925613, 4.13367633193593308154224536720, 5.98083503443993720406932585882, 7.76250669663305613210077564119, 8.967593162309253346710299641169, 9.809514923245685914950115524611, 11.82226239066862034035907886284, 13.51901272130864475680765935145