| L(s) = 1 | + 9.85e16·2-s − 3.53e26·3-s + 7.11e33·4-s + 7.51e38·5-s − 3.48e43·6-s − 5.05e46·7-s + 4.45e50·8-s + 3.37e52·9-s + 7.40e55·10-s + 1.01e58·11-s − 2.51e60·12-s − 2.38e61·13-s − 4.97e63·14-s − 2.65e65·15-s + 2.54e67·16-s + 2.09e67·17-s + 3.32e69·18-s − 7.22e69·19-s + 5.34e72·20-s + 1.78e73·21-s + 1.00e75·22-s − 1.40e75·23-s − 1.57e77·24-s + 1.79e77·25-s − 2.34e78·26-s + 2.03e79·27-s − 3.59e80·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 1.17·3-s + 2.74·4-s + 1.21·5-s − 2.26·6-s − 0.631·7-s + 3.36·8-s + 0.369·9-s + 2.34·10-s + 1.61·11-s − 3.20·12-s − 0.357·13-s − 1.22·14-s − 1.41·15-s + 3.77·16-s + 0.107·17-s + 0.715·18-s − 0.0772·19-s + 3.31·20-s + 0.739·21-s + 3.13·22-s − 0.373·23-s − 3.94·24-s + 0.465·25-s − 0.691·26-s + 0.737·27-s − 1.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(112-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(56)\) |
\(\approx\) |
\(7.157539737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.157539737\) |
| \(L(\frac{113}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 9.85e16T + 2.59e33T^{2} \) |
| 3 | \( 1 + 3.53e26T + 9.12e52T^{2} \) |
| 5 | \( 1 - 7.51e38T + 3.85e77T^{2} \) |
| 7 | \( 1 + 5.05e46T + 6.39e93T^{2} \) |
| 11 | \( 1 - 1.01e58T + 3.93e115T^{2} \) |
| 13 | \( 1 + 2.38e61T + 4.44e123T^{2} \) |
| 17 | \( 1 - 2.09e67T + 3.80e136T^{2} \) |
| 19 | \( 1 + 7.22e69T + 8.74e141T^{2} \) |
| 23 | \( 1 + 1.40e75T + 1.41e151T^{2} \) |
| 29 | \( 1 + 2.96e80T + 2.11e162T^{2} \) |
| 31 | \( 1 - 6.06e82T + 3.47e165T^{2} \) |
| 37 | \( 1 - 9.68e86T + 1.17e174T^{2} \) |
| 41 | \( 1 - 3.73e89T + 1.04e179T^{2} \) |
| 43 | \( 1 + 8.35e90T + 2.06e181T^{2} \) |
| 47 | \( 1 - 4.25e92T + 4.00e185T^{2} \) |
| 53 | \( 1 - 6.81e95T + 2.48e191T^{2} \) |
| 59 | \( 1 - 2.04e98T + 3.66e196T^{2} \) |
| 61 | \( 1 - 1.55e98T + 1.48e198T^{2} \) |
| 67 | \( 1 - 1.19e101T + 4.94e202T^{2} \) |
| 71 | \( 1 - 4.59e102T + 3.08e205T^{2} \) |
| 73 | \( 1 - 1.35e103T + 6.74e206T^{2} \) |
| 79 | \( 1 + 1.69e105T + 4.33e210T^{2} \) |
| 83 | \( 1 - 3.18e106T + 1.04e213T^{2} \) |
| 89 | \( 1 + 1.15e108T + 2.41e216T^{2} \) |
| 97 | \( 1 - 2.20e110T + 3.40e220T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.55792474594279613813712117071, −12.33295794202538978977146936554, −11.42482760684825916766312272956, −9.957749051016018682314887126823, −6.66797540660773365743328258070, −6.17279805371584202033844471241, −5.23499518872372031228170950252, −3.95916409607312310628341229452, −2.47958667889711044172364999774, −1.18634445197179777175971699624,
1.18634445197179777175971699624, 2.47958667889711044172364999774, 3.95916409607312310628341229452, 5.23499518872372031228170950252, 6.17279805371584202033844471241, 6.66797540660773365743328258070, 9.957749051016018682314887126823, 11.42482760684825916766312272956, 12.33295794202538978977146936554, 13.55792474594279613813712117071
