| L(s) = 1 | − 5.83e14·2-s − 6.70e22·3-s + 1.82e29·4-s + 3.35e33·5-s + 3.91e37·6-s + 6.90e40·7-s − 1.39e43·8-s − 1.45e46·9-s − 1.95e48·10-s − 7.70e49·11-s − 1.22e52·12-s + 2.16e53·13-s − 4.03e55·14-s − 2.24e56·15-s − 2.07e58·16-s − 6.79e59·17-s + 8.52e60·18-s + 6.40e61·19-s + 6.11e62·20-s − 4.62e63·21-s + 4.49e64·22-s + 8.09e65·23-s + 9.37e65·24-s − 5.18e67·25-s − 1.26e68·26-s + 2.25e69·27-s + 1.25e70·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.485·3-s + 1.15·4-s + 0.421·5-s + 0.711·6-s + 0.710·7-s − 0.221·8-s − 0.764·9-s − 0.618·10-s − 0.239·11-s − 0.558·12-s + 0.203·13-s − 1.04·14-s − 0.204·15-s − 0.825·16-s − 1.43·17-s + 1.12·18-s + 0.612·19-s + 0.485·20-s − 0.344·21-s + 0.351·22-s + 0.732·23-s + 0.107·24-s − 0.822·25-s − 0.298·26-s + 0.856·27-s + 0.818·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(98-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+97/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(49)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{99}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 5.83e14T + 1.58e29T^{2} \) |
| 3 | \( 1 + 6.70e22T + 1.90e46T^{2} \) |
| 5 | \( 1 - 3.35e33T + 6.31e67T^{2} \) |
| 7 | \( 1 - 6.90e40T + 9.42e81T^{2} \) |
| 11 | \( 1 + 7.70e49T + 1.03e101T^{2} \) |
| 13 | \( 1 - 2.16e53T + 1.12e108T^{2} \) |
| 17 | \( 1 + 6.79e59T + 2.25e119T^{2} \) |
| 19 | \( 1 - 6.40e61T + 1.09e124T^{2} \) |
| 23 | \( 1 - 8.09e65T + 1.22e132T^{2} \) |
| 29 | \( 1 - 4.54e70T + 7.12e141T^{2} \) |
| 31 | \( 1 - 3.94e72T + 4.59e144T^{2} \) |
| 37 | \( 1 + 1.34e76T + 1.30e152T^{2} \) |
| 41 | \( 1 + 1.03e77T + 2.75e156T^{2} \) |
| 43 | \( 1 - 3.26e79T + 2.79e158T^{2} \) |
| 47 | \( 1 + 1.05e81T + 1.56e162T^{2} \) |
| 53 | \( 1 + 3.91e83T + 1.79e167T^{2} \) |
| 59 | \( 1 - 8.07e85T + 5.92e171T^{2} \) |
| 61 | \( 1 + 4.86e86T + 1.50e173T^{2} \) |
| 67 | \( 1 + 6.80e87T + 1.34e177T^{2} \) |
| 71 | \( 1 + 4.42e89T + 3.73e179T^{2} \) |
| 73 | \( 1 - 2.87e90T + 5.52e180T^{2} \) |
| 79 | \( 1 - 4.78e91T + 1.17e184T^{2} \) |
| 83 | \( 1 + 4.25e89T + 1.41e186T^{2} \) |
| 89 | \( 1 - 5.83e94T + 1.23e189T^{2} \) |
| 97 | \( 1 - 6.44e95T + 5.21e192T^{2} \) |
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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
−13.78998639941535483776185339672, −11.56010598176260962082909051251, −10.58804703115399438569616011468, −9.115006586813630985134681570722, −8.028438225777052213417312371199, −6.46137395932612243221285009549, −4.86518407613962908558218745704, −2.45623938634648394633120001447, −1.17230309551921326041589819226, 0,
1.17230309551921326041589819226, 2.45623938634648394633120001447, 4.86518407613962908558218745704, 6.46137395932612243221285009549, 8.028438225777052213417312371199, 9.115006586813630985134681570722, 10.58804703115399438569616011468, 11.56010598176260962082909051251, 13.78998639941535483776185339672
