| L(s) = 1 | + 3.50·2-s + 4.27·4-s − 15.3·5-s − 7·7-s − 13.0·8-s − 53.6·10-s − 11·11-s + 5.90·13-s − 24.5·14-s − 79.9·16-s + 73.2·17-s + 80.3·19-s − 65.4·20-s − 38.5·22-s + 190.·23-s + 109.·25-s + 20.7·26-s − 29.9·28-s + 285.·29-s + 85.3·31-s − 175.·32-s + 256.·34-s + 107.·35-s − 149.·37-s + 281.·38-s + 199.·40-s − 13.9·41-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.534·4-s − 1.36·5-s − 0.377·7-s − 0.576·8-s − 1.69·10-s − 0.301·11-s + 0.126·13-s − 0.468·14-s − 1.24·16-s + 1.04·17-s + 0.970·19-s − 0.731·20-s − 0.373·22-s + 1.72·23-s + 0.874·25-s + 0.156·26-s − 0.201·28-s + 1.82·29-s + 0.494·31-s − 0.970·32-s + 1.29·34-s + 0.517·35-s − 0.662·37-s + 1.20·38-s + 0.789·40-s − 0.0530·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.434286352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434286352\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.50T + 8T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 13 | \( 1 - 5.90T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 85.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 13.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 148.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 94.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 189.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 356.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 655.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 110.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 372.T + 9.12e5T^{2} \) |
| 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
−10.24022825879526835936040370793, −9.121663593155540472888652924189, −8.191935624637423795010025924167, −7.29288316017452578450057342191, −6.41105163973568419056448121556, −5.21988589815245406818198945664, −4.59160597174282663975689284017, −3.40861525127283422849689813249, −3.03748069737443641897987716967, −0.74723587190876114952230456649,
0.74723587190876114952230456649, 3.03748069737443641897987716967, 3.40861525127283422849689813249, 4.59160597174282663975689284017, 5.21988589815245406818198945664, 6.41105163973568419056448121556, 7.29288316017452578450057342191, 8.191935624637423795010025924167, 9.121663593155540472888652924189, 10.24022825879526835936040370793
