| L(s) = 1 | + 4.40·2-s + 11.3·4-s + 3.94·5-s − 14.1·7-s + 14.8·8-s + 17.3·10-s + 4.36·11-s + 37.8·13-s − 62.1·14-s − 25.6·16-s − 86.4·17-s + 87.3·19-s + 44.8·20-s + 19.2·22-s − 116.·23-s − 109.·25-s + 166.·26-s − 160.·28-s + 78.5·29-s − 28.5·31-s − 231.·32-s − 380.·34-s − 55.6·35-s − 90.6·37-s + 384.·38-s + 58.4·40-s + 69.2·41-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.42·4-s + 0.352·5-s − 0.762·7-s + 0.655·8-s + 0.548·10-s + 0.119·11-s + 0.807·13-s − 1.18·14-s − 0.401·16-s − 1.23·17-s + 1.05·19-s + 0.501·20-s + 0.186·22-s − 1.06·23-s − 0.875·25-s + 1.25·26-s − 1.08·28-s + 0.503·29-s − 0.165·31-s − 1.27·32-s − 1.91·34-s − 0.268·35-s − 0.402·37-s + 1.64·38-s + 0.231·40-s + 0.263·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 4.40T + 8T^{2} \) |
| 5 | \( 1 - 3.94T + 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 - 4.36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 90.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 329.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 65.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 337.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 65.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 556.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 850.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 318.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 939.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 78.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 771.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 633.T + 9.12e5T^{2} \) |
| 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
−8.293814430153541730570536654729, −7.15434951814211590828103136636, −6.41458608050209023227849941062, −5.94920724759976629421838129818, −5.13298712540771209673083787063, −4.18204647126844475321196969825, −3.54153975869664364252465763036, −2.68763866340234213287480769120, −1.67187343984148581106217454646, 0,
1.67187343984148581106217454646, 2.68763866340234213287480769120, 3.54153975869664364252465763036, 4.18204647126844475321196969825, 5.13298712540771209673083787063, 5.94920724759976629421838129818, 6.41458608050209023227849941062, 7.15434951814211590828103136636, 8.293814430153541730570536654729
