| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 0.414·7-s + 4.41·8-s + 11-s + 2.82·13-s − 0.999·14-s + 2.99·16-s + 2.41·17-s + 6.41·19-s + 2.41·22-s + 23-s + 6.82·26-s − 1.58·28-s − 1.17·29-s − 8.48·31-s − 1.58·32-s + 5.82·34-s + 0.171·37-s + 15.4·38-s + 10.8·41-s + 11.6·43-s + 3.82·44-s + 2.41·46-s − 7.48·47-s − 6.82·49-s + 10.8·52-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.156·7-s + 1.56·8-s + 0.301·11-s + 0.784·13-s − 0.267·14-s + 0.749·16-s + 0.585·17-s + 1.47·19-s + 0.514·22-s + 0.208·23-s + 1.33·26-s − 0.299·28-s − 0.217·29-s − 1.52·31-s − 0.280·32-s + 0.999·34-s + 0.0282·37-s + 2.51·38-s + 1.70·41-s + 1.77·43-s + 0.577·44-s + 0.355·46-s − 1.09·47-s − 0.975·49-s + 1.50·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.432303933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.432303933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.171T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 0.343T + 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
| 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
−9.042108873220691970721145327688, −7.78903571165956974370921895178, −7.21133434994057695213254037416, −6.28788355886798183937497448502, −5.67279215383912692861881256720, −5.05311411854726103041484059790, −3.98889211675823804481276091346, −3.47813171042685847801851810315, −2.58177776637967524473544864204, −1.29914144667592596561329551750,
1.29914144667592596561329551750, 2.58177776637967524473544864204, 3.47813171042685847801851810315, 3.98889211675823804481276091346, 5.05311411854726103041484059790, 5.67279215383912692861881256720, 6.28788355886798183937497448502, 7.21133434994057695213254037416, 7.78903571165956974370921895178, 9.042108873220691970721145327688
