| L(s) = 1 | − 2.79·2-s − 0.185·4-s − 7.18·5-s + 19.9·7-s + 22.8·8-s + 20.0·10-s + 20.2·11-s − 69.3·13-s − 55.6·14-s − 62.4·16-s − 17·17-s − 2.16·19-s + 1.33·20-s − 56.7·22-s + 13.0·23-s − 73.3·25-s + 193.·26-s − 3.69·28-s − 246.·29-s − 166.·31-s − 8.39·32-s + 47.5·34-s − 143.·35-s − 162.·37-s + 6.04·38-s − 164.·40-s − 253.·41-s + ⋯ |
| L(s) = 1 | − 0.988·2-s − 0.0231·4-s − 0.642·5-s + 1.07·7-s + 1.01·8-s + 0.635·10-s + 0.555·11-s − 1.47·13-s − 1.06·14-s − 0.976·16-s − 0.242·17-s − 0.0261·19-s + 0.0149·20-s − 0.549·22-s + 0.118·23-s − 0.586·25-s + 1.46·26-s − 0.0249·28-s − 1.58·29-s − 0.966·31-s − 0.0463·32-s + 0.239·34-s − 0.691·35-s − 0.720·37-s + 0.0258·38-s − 0.649·40-s − 0.965·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 2.79T + 8T^{2} \) |
| 5 | \( 1 + 7.18T + 125T^{2} \) |
| 7 | \( 1 - 19.9T + 343T^{2} \) |
| 11 | \( 1 - 20.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 69.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 2.16T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 556.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 332.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 169.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 509.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 485.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
| 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
−11.70828352468553040967900413749, −10.96965997318068539077218433579, −9.778701373800632889397996087190, −8.897996537578487955617633066410, −7.83474739392216022566928054928, −7.22694930000157491336706397957, −5.17720788170549705638842089984, −4.08902997194734833753093184761, −1.80764492040638991603309607579, 0,
1.80764492040638991603309607579, 4.08902997194734833753093184761, 5.17720788170549705638842089984, 7.22694930000157491336706397957, 7.83474739392216022566928054928, 8.897996537578487955617633066410, 9.778701373800632889397996087190, 10.96965997318068539077218433579, 11.70828352468553040967900413749
