| L(s) = 1 | − 5.49·2-s + 3.98·3-s + 22.1·4-s − 5·5-s − 21.8·6-s + 23.4·7-s − 77.7·8-s − 11.1·9-s + 27.4·10-s − 11·11-s + 88.3·12-s − 8.78·13-s − 128.·14-s − 19.9·15-s + 249.·16-s − 67.5·17-s + 61.0·18-s + 19·19-s − 110.·20-s + 93.5·21-s + 60.4·22-s + 110.·23-s − 309.·24-s + 25·25-s + 48.2·26-s − 151.·27-s + 520.·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.766·3-s + 2.77·4-s − 0.447·5-s − 1.48·6-s + 1.26·7-s − 3.43·8-s − 0.411·9-s + 0.868·10-s − 0.301·11-s + 2.12·12-s − 0.187·13-s − 2.46·14-s − 0.342·15-s + 3.90·16-s − 0.963·17-s + 0.799·18-s + 0.229·19-s − 1.23·20-s + 0.972·21-s + 0.585·22-s + 0.999·23-s − 2.63·24-s + 0.200·25-s + 0.364·26-s − 1.08·27-s + 3.51·28-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 | \( 1 + 5.49T + 8T^{2} \) |
| 3 | \( 1 - 3.98T + 27T^{2} \) |
| 7 | \( 1 - 23.4T + 343T^{2} \) |
| 13 | \( 1 + 8.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 21.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 527.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 63.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 787.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 337.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 676.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 669.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
| 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
−8.986900141281443318108156561024, −8.336613664776332670983564044861, −7.83497700009678223022794654556, −7.19285840627876209388974701195, −6.09523113868482486564227989706, −4.76199709871080112403264244944, −3.13095148977499255323388510372, −2.31616347005034088092043167841, −1.30951308323370640023036175248, 0,
1.30951308323370640023036175248, 2.31616347005034088092043167841, 3.13095148977499255323388510372, 4.76199709871080112403264244944, 6.09523113868482486564227989706, 7.19285840627876209388974701195, 7.83497700009678223022794654556, 8.336613664776332670983564044861, 8.986900141281443318108156561024
