| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 3·9-s + 2·10-s + 2·11-s + 2·13-s − 4·16-s − 3·17-s − 6·18-s + 2·19-s + 2·20-s + 4·22-s + 23-s + 25-s + 4·26-s + 7·29-s + 5·31-s − 8·32-s − 6·34-s − 6·36-s + 11·37-s + 4·38-s − 41-s + 4·44-s − 3·45-s + 2·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 9-s + 0.632·10-s + 0.603·11-s + 0.554·13-s − 16-s − 0.727·17-s − 1.41·18-s + 0.458·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s + 1.29·29-s + 0.898·31-s − 1.41·32-s − 1.02·34-s − 36-s + 1.80·37-s + 0.648·38-s − 0.156·41-s + 0.603·44-s − 0.447·45-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.501230311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.501230311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.263970881296066983490176011043, −6.99788152624213115553514070657, −6.45697356274813578150694386716, −5.87176551907712567991921014352, −5.27308193075008769160572382867, −4.45846692453006168600746685551, −3.82026530899319001848621226796, −2.86980424199587493895479247872, −2.38005066461420004928870391491, −0.921910412570228599339074523899,
0.921910412570228599339074523899, 2.38005066461420004928870391491, 2.86980424199587493895479247872, 3.82026530899319001848621226796, 4.45846692453006168600746685551, 5.27308193075008769160572382867, 5.87176551907712567991921014352, 6.45697356274813578150694386716, 6.99788152624213115553514070657, 8.263970881296066983490176011043
