| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 3·18-s + 20-s + 4·22-s + 25-s + 6·26-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s + 40-s − 2·41-s + 4·43-s + 4·44-s − 3·45-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.447·45-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369641784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369641784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−11.21735898278523397026755152007, −10.27331483782898950038120178987, −8.943481127314352432901190185250, −8.517395214779930601558675802260, −6.94587741637952917868946998696, −6.19981958164339530240516278001, −5.43748531590055688874527891076, −4.08647900838425361258038365025, −3.14888451818473432527395410089, −1.60311130730382042811195419884,
1.60311130730382042811195419884, 3.14888451818473432527395410089, 4.08647900838425361258038365025, 5.43748531590055688874527891076, 6.19981958164339530240516278001, 6.94587741637952917868946998696, 8.517395214779930601558675802260, 8.943481127314352432901190185250, 10.27331483782898950038120178987, 11.21735898278523397026755152007
