L(s) = 1 | + 2-s + 4-s + 1.73·5-s + 7-s + 8-s + 1.73·10-s + 1.26·11-s − 13-s + 14-s + 16-s − 0.464·17-s + 4.19·19-s + 1.73·20-s + 1.26·22-s + 4.73·23-s − 2.00·25-s − 26-s + 28-s − 0.464·29-s − 6.19·31-s + 32-s − 0.464·34-s + 1.73·35-s + 7.19·37-s + 4.19·38-s + 1.73·40-s + 9.46·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.774·5-s + 0.377·7-s + 0.353·8-s + 0.547·10-s + 0.382·11-s − 0.277·13-s + 0.267·14-s + 0.250·16-s − 0.112·17-s + 0.962·19-s + 0.387·20-s + 0.270·22-s + 0.986·23-s − 0.400·25-s − 0.196·26-s + 0.188·28-s − 0.0861·29-s − 1.11·31-s + 0.176·32-s − 0.0795·34-s + 0.292·35-s + 1.18·37-s + 0.680·38-s + 0.273·40-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.097753181\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.097753181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.708802112200545054979205940372, −9.238895208381688891653603629952, −8.015974409226721273100408599541, −7.20639169902003093823594447526, −6.30596673154967438129368524724, −5.48346209543955450969979721753, −4.76615008587015873339604783187, −3.63900365053701948602781136089, −2.53876074813396339084514628444, −1.41157427804735858984491658217,
1.41157427804735858984491658217, 2.53876074813396339084514628444, 3.63900365053701948602781136089, 4.76615008587015873339604783187, 5.48346209543955450969979721753, 6.30596673154967438129368524724, 7.20639169902003093823594447526, 8.015974409226721273100408599541, 9.238895208381688891653603629952, 9.708802112200545054979205940372