| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s + 4·6-s + 31·7-s + 8·8-s − 23·9-s + 57·11-s + 8·12-s + 52·13-s + 62·14-s + 16·16-s − 69·17-s − 46·18-s + 19·19-s + 62·21-s + 114·22-s + 72·23-s + 16·24-s + 104·26-s − 100·27-s + 124·28-s − 150·29-s + 32·31-s + 32·32-s + 114·33-s − 138·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.272·6-s + 1.67·7-s + 0.353·8-s − 0.851·9-s + 1.56·11-s + 0.192·12-s + 1.10·13-s + 1.18·14-s + 1/4·16-s − 0.984·17-s − 0.602·18-s + 0.229·19-s + 0.644·21-s + 1.10·22-s + 0.652·23-s + 0.136·24-s + 0.784·26-s − 0.712·27-s + 0.836·28-s − 0.960·29-s + 0.185·31-s + 0.176·32-s + 0.601·33-s − 0.696·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.104100947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.104100947\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 31 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 69 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 258 T + p^{3} T^{2} \) |
| 43 | \( 1 - 67 T + p^{3} T^{2} \) |
| 47 | \( 1 + 579 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 330 T + p^{3} T^{2} \) |
| 61 | \( 1 + 13 T + p^{3} T^{2} \) |
| 67 | \( 1 - 856 T + p^{3} T^{2} \) |
| 71 | \( 1 - 642 T + p^{3} T^{2} \) |
| 73 | \( 1 - 487 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1424 T + p^{3} T^{2} \) |
| 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
−9.468403945590368205110897873235, −8.628509134903463475400104975273, −8.156975265544675343964682168438, −6.99878066250355082865665439306, −6.13565830942084104388641767092, −5.19262497592221792044214395202, −4.27269192310621197493886340231, −3.46996178236121655010890473729, −2.11670278387815895770322109818, −1.21926936824500023097318768445,
1.21926936824500023097318768445, 2.11670278387815895770322109818, 3.46996178236121655010890473729, 4.27269192310621197493886340231, 5.19262497592221792044214395202, 6.13565830942084104388641767092, 6.99878066250355082865665439306, 8.156975265544675343964682168438, 8.628509134903463475400104975273, 9.468403945590368205110897873235
