| L(s) = 1 | − 5.47·2-s − 2.61·3-s + 22.0·4-s + 5·5-s + 14.3·6-s − 5.28·7-s − 76.8·8-s − 20.1·9-s − 27.3·10-s + 11·11-s − 57.6·12-s − 39.7·13-s + 28.9·14-s − 13.0·15-s + 244.·16-s + 64.4·17-s + 110.·18-s + 19·19-s + 110.·20-s + 13.8·21-s − 60.2·22-s − 191.·23-s + 200.·24-s + 25·25-s + 217.·26-s + 123.·27-s − 116.·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.503·3-s + 2.75·4-s + 0.447·5-s + 0.975·6-s − 0.285·7-s − 3.39·8-s − 0.746·9-s − 0.866·10-s + 0.301·11-s − 1.38·12-s − 0.848·13-s + 0.553·14-s − 0.225·15-s + 3.82·16-s + 0.920·17-s + 1.44·18-s + 0.229·19-s + 1.23·20-s + 0.143·21-s − 0.584·22-s − 1.73·23-s + 1.70·24-s + 0.200·25-s + 1.64·26-s + 0.879·27-s − 0.785·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 5.47T + 8T^{2} \) |
| 3 | \( 1 + 2.61T + 27T^{2} \) |
| 7 | \( 1 + 5.28T + 343T^{2} \) |
| 13 | \( 1 + 39.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 237.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 505.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 450.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 196.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 80.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 897.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
| 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
−9.341577518926505630981002938291, −8.164497926754872455089582224386, −7.916537385170212743276735170569, −6.42714665343532707178747777407, −6.37667578248113281434354226172, −5.11999560379559736781489319516, −3.16845858400665764203651744784, −2.26285939762291746931162964867, −1.03551601378853316931698247308, 0,
1.03551601378853316931698247308, 2.26285939762291746931162964867, 3.16845858400665764203651744784, 5.11999560379559736781489319516, 6.37667578248113281434354226172, 6.42714665343532707178747777407, 7.916537385170212743276735170569, 8.164497926754872455089582224386, 9.341577518926505630981002938291
