| L(s) = 1 | + 4·2-s + 11·3-s + 16·4-s + 44·6-s + 142·7-s + 64·8-s − 122·9-s + 777·11-s + 176·12-s − 884·13-s + 568·14-s + 256·16-s + 27·17-s − 488·18-s + 1.14e3·19-s + 1.56e3·21-s + 3.10e3·22-s − 1.85e3·23-s + 704·24-s − 3.53e3·26-s − 4.01e3·27-s + 2.27e3·28-s − 4.92e3·29-s + 1.80e3·31-s + 1.02e3·32-s + 8.54e3·33-s + 108·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.705·3-s + 1/2·4-s + 0.498·6-s + 1.09·7-s + 0.353·8-s − 0.502·9-s + 1.93·11-s + 0.352·12-s − 1.45·13-s + 0.774·14-s + 1/4·16-s + 0.0226·17-s − 0.355·18-s + 0.727·19-s + 0.772·21-s + 1.36·22-s − 0.730·23-s + 0.249·24-s − 1.02·26-s − 1.05·27-s + 0.547·28-s − 1.08·29-s + 0.336·31-s + 0.176·32-s + 1.36·33-s + 0.0160·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.269504272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.269504272\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 11 T + p^{5} T^{2} \) |
| 7 | \( 1 - 142 T + p^{5} T^{2} \) |
| 11 | \( 1 - 777 T + p^{5} T^{2} \) |
| 13 | \( 1 + 68 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 27 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1145 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1854 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4920 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1802 T + p^{5} T^{2} \) |
| 37 | \( 1 + 13178 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15123 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7844 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6732 T + p^{5} T^{2} \) |
| 53 | \( 1 + 3414 T + p^{5} T^{2} \) |
| 59 | \( 1 - 33960 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47402 T + p^{5} T^{2} \) |
| 67 | \( 1 - 13177 T + p^{5} T^{2} \) |
| 71 | \( 1 + 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 - 59821 T + p^{5} T^{2} \) |
| 79 | \( 1 - 75830 T + p^{5} T^{2} \) |
| 83 | \( 1 + 46299 T + p^{5} T^{2} \) |
| 89 | \( 1 + 30585 T + p^{5} T^{2} \) |
| 97 | \( 1 + 104018 T + p^{5} T^{2} \) |
| 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
−14.43327742114753768288761060165, −13.86969335254632821712234970471, −12.08231143651369480790503183433, −11.48626035002091776530861244625, −9.655919258705577388425876980814, −8.352181765215564025430748402341, −7.00841065172353884478348561074, −5.22894072911208369526589067405, −3.71791922551417773450983457539, −1.91787968297563495959594049953,
1.91787968297563495959594049953, 3.71791922551417773450983457539, 5.22894072911208369526589067405, 7.00841065172353884478348561074, 8.352181765215564025430748402341, 9.655919258705577388425876980814, 11.48626035002091776530861244625, 12.08231143651369480790503183433, 13.86969335254632821712234970471, 14.43327742114753768288761060165
