| L(s) = 1 | + 3.44e17·2-s + 1.64e27·3-s + 7.68e34·4-s + 3.84e38·5-s + 5.64e44·6-s + 3.35e48·7-s + 1.21e52·8-s − 4.70e54·9-s + 1.32e56·10-s + 1.11e60·11-s + 1.26e62·12-s − 2.88e63·13-s + 1.15e66·14-s + 6.30e65·15-s + 9.88e68·16-s − 4.73e69·17-s − 1.61e72·18-s + 5.83e73·19-s + 2.95e73·20-s + 5.50e75·21-s + 3.84e77·22-s + 1.71e78·23-s + 1.99e79·24-s − 2.40e80·25-s − 9.94e80·26-s − 1.98e82·27-s + 2.57e83·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.603·3-s + 1.85·4-s + 0.0247·5-s + 1.01·6-s + 0.855·7-s + 1.43·8-s − 0.635·9-s + 0.0417·10-s + 1.47·11-s + 1.11·12-s − 0.256·13-s + 1.44·14-s + 0.0149·15-s + 0.572·16-s − 0.0841·17-s − 1.07·18-s + 1.72·19-s + 0.0457·20-s + 0.516·21-s + 2.48·22-s + 0.861·23-s + 0.866·24-s − 0.999·25-s − 0.432·26-s − 0.987·27-s + 1.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(116-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(58)\) |
\(\approx\) |
\(9.126525905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.126525905\) |
| \(L(\frac{117}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.44e17T + 4.15e34T^{2} \) |
| 3 | \( 1 - 1.64e27T + 7.39e54T^{2} \) |
| 5 | \( 1 - 3.84e38T + 2.40e80T^{2} \) |
| 7 | \( 1 - 3.35e48T + 1.53e97T^{2} \) |
| 11 | \( 1 - 1.11e60T + 5.75e119T^{2} \) |
| 13 | \( 1 + 2.88e63T + 1.26e128T^{2} \) |
| 17 | \( 1 + 4.73e69T + 3.17e141T^{2} \) |
| 19 | \( 1 - 5.83e73T + 1.13e147T^{2} \) |
| 23 | \( 1 - 1.71e78T + 3.96e156T^{2} \) |
| 29 | \( 1 - 1.92e84T + 1.49e168T^{2} \) |
| 31 | \( 1 + 1.36e85T + 3.21e171T^{2} \) |
| 37 | \( 1 + 8.56e89T + 2.20e180T^{2} \) |
| 41 | \( 1 - 1.42e92T + 2.95e185T^{2} \) |
| 43 | \( 1 - 1.19e94T + 7.06e187T^{2} \) |
| 47 | \( 1 - 2.47e96T + 1.95e192T^{2} \) |
| 53 | \( 1 + 1.09e99T + 1.95e198T^{2} \) |
| 59 | \( 1 + 1.22e102T + 4.44e203T^{2} \) |
| 61 | \( 1 + 1.14e102T + 2.05e205T^{2} \) |
| 67 | \( 1 - 1.02e105T + 9.96e209T^{2} \) |
| 71 | \( 1 + 1.97e106T + 7.84e212T^{2} \) |
| 73 | \( 1 - 1.60e106T + 1.91e214T^{2} \) |
| 79 | \( 1 + 1.30e109T + 1.68e218T^{2} \) |
| 83 | \( 1 - 5.66e109T + 4.94e220T^{2} \) |
| 89 | \( 1 + 8.44e111T + 1.51e224T^{2} \) |
| 97 | \( 1 - 2.99e114T + 3.01e228T^{2} \) |
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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
−14.00817854881250599797137021808, −12.13989677549476709456884513451, −11.32040629999597703809485228038, −9.096848683105834912504769133622, −7.44047239253391616132939743615, −5.98606680045234834364961235762, −4.80740988275090048425508743545, −3.65531772663177755828582839486, −2.63559626348029599304043051512, −1.31419440380015639161310318623,
1.31419440380015639161310318623, 2.63559626348029599304043051512, 3.65531772663177755828582839486, 4.80740988275090048425508743545, 5.98606680045234834364961235762, 7.44047239253391616132939743615, 9.096848683105834912504769133622, 11.32040629999597703809485228038, 12.13989677549476709456884513451, 14.00817854881250599797137021808
