L(s) = 1 | + 2.23·2-s − 3-s + 3.00·4-s − 5-s − 2.23·6-s + 7-s + 2.23·8-s + 9-s − 2.23·10-s − 2.47·11-s − 3.00·12-s − 4.47·13-s + 2.23·14-s + 15-s − 0.999·16-s − 2·17-s + 2.23·18-s + 6.47·19-s − 3.00·20-s − 21-s − 5.52·22-s + 4·23-s − 2.23·24-s + 25-s − 10.0·26-s − 27-s + 3.00·28-s + ⋯ |
L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.447·5-s − 0.912·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.707·10-s − 0.745·11-s − 0.866·12-s − 1.24·13-s + 0.597·14-s + 0.258·15-s − 0.249·16-s − 0.485·17-s + 0.527·18-s + 1.48·19-s − 0.670·20-s − 0.218·21-s − 1.17·22-s + 0.834·23-s − 0.456·24-s + 0.200·25-s − 1.96·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.731878856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731878856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 0.944T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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
−13.64300893606418748618108877373, −12.78164098571730908744395799935, −11.82381989334064317755500600093, −11.22830280592174644561216623642, −9.782036460362608195092526135262, −7.82513692382023103813259447991, −6.69481828136884877767437788201, −5.25936024831167293412664975803, −4.60463390503738808095240317214, −2.89131068890819257465193141069,
2.89131068890819257465193141069, 4.60463390503738808095240317214, 5.25936024831167293412664975803, 6.69481828136884877767437788201, 7.82513692382023103813259447991, 9.782036460362608195092526135262, 11.22830280592174644561216623642, 11.82381989334064317755500600093, 12.78164098571730908744395799935, 13.64300893606418748618108877373