| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 25·5-s − 36·6-s + 135.·7-s + 64·8-s + 81·9-s − 100·10-s − 130.·11-s − 144·12-s − 368.·13-s + 543.·14-s + 225·15-s + 256·16-s − 1.06e3·17-s + 324·18-s + 361·19-s − 400·20-s − 1.22e3·21-s − 521.·22-s + 244.·23-s − 576·24-s + 625·25-s − 1.47e3·26-s − 729·27-s + 2.17e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.04·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.325·11-s − 0.288·12-s − 0.605·13-s + 0.741·14-s + 0.258·15-s + 0.250·16-s − 0.890·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.605·21-s − 0.229·22-s + 0.0963·23-s − 0.204·24-s + 0.200·25-s − 0.427·26-s − 0.192·27-s + 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 130.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 368.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.06e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 244.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.46e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.97e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.27e5T + 8.58e9T^{2} \) |
| 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
−9.689337666653168254387224897260, −8.410904084402128346589445876530, −7.57706263626483443821754486059, −6.75966199641174939882463065849, −5.59970923122564340070553225093, −4.81658872554686109571871446200, −4.10436063028173490617485186255, −2.66622879837888642486152561807, −1.47280329067203844904045629101, 0,
1.47280329067203844904045629101, 2.66622879837888642486152561807, 4.10436063028173490617485186255, 4.81658872554686109571871446200, 5.59970923122564340070553225093, 6.75966199641174939882463065849, 7.57706263626483443821754486059, 8.410904084402128346589445876530, 9.689337666653168254387224897260
