| L(s) = 1 | − 1.27·2-s + 2.56·3-s − 0.385·4-s − 3.26·6-s − 0.840·7-s + 3.03·8-s + 3.59·9-s − 4.34·11-s − 0.989·12-s + 5.35·13-s + 1.06·14-s − 3.08·16-s + 3.54·17-s − 4.56·18-s + 1.48·19-s − 2.15·21-s + 5.51·22-s + 8.79·23-s + 7.78·24-s − 6.80·26-s + 1.51·27-s + 0.323·28-s + 0.249·29-s − 0.752·31-s − 2.14·32-s − 11.1·33-s − 4.51·34-s + ⋯ |
| L(s) = 1 | − 0.898·2-s + 1.48·3-s − 0.192·4-s − 1.33·6-s − 0.317·7-s + 1.07·8-s + 1.19·9-s − 1.30·11-s − 0.285·12-s + 1.48·13-s + 0.285·14-s − 0.770·16-s + 0.860·17-s − 1.07·18-s + 0.341·19-s − 0.471·21-s + 1.17·22-s + 1.83·23-s + 1.58·24-s − 1.33·26-s + 0.292·27-s + 0.0612·28-s + 0.0464·29-s − 0.135·31-s − 0.379·32-s − 1.93·33-s − 0.773·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914166772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914166772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 0.840T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 - 0.249T + 29T^{2} \) |
| 31 | \( 1 + 0.752T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 - 3.98T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 0.0288T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 + 0.952T + 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
| 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
−8.298472834367433621894198391505, −7.80882773345829439193331938267, −7.35228326953746088520764288740, −6.25244672474383695877286476599, −5.24705590509381494206940853666, −4.42702610795737497904761136294, −3.33598556422993347524642182686, −3.01125962779622659865521299960, −1.79940546167011464532532530504, −0.848483247052846645925874884363,
0.848483247052846645925874884363, 1.79940546167011464532532530504, 3.01125962779622659865521299960, 3.33598556422993347524642182686, 4.42702610795737497904761136294, 5.24705590509381494206940853666, 6.25244672474383695877286476599, 7.35228326953746088520764288740, 7.80882773345829439193331938267, 8.298472834367433621894198391505
