| L(s) = 1 | + 3·3-s + 16.6·5-s + 9·9-s + 71.7·11-s + 65.3·13-s + 49.9·15-s − 90.4·17-s + 163.·19-s − 79.2·23-s + 152.·25-s + 27·27-s − 43.2·29-s + 135.·31-s + 215.·33-s + 270.·37-s + 196.·39-s − 152.·41-s + 177.·43-s + 149.·45-s − 45.6·47-s − 271.·51-s − 158.·53-s + 1.19e3·55-s + 491.·57-s − 391.·59-s − 551.·61-s + 1.08e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.48·5-s + 0.333·9-s + 1.96·11-s + 1.39·13-s + 0.860·15-s − 1.29·17-s + 1.97·19-s − 0.718·23-s + 1.21·25-s + 0.192·27-s − 0.277·29-s + 0.785·31-s + 1.13·33-s + 1.20·37-s + 0.805·39-s − 0.579·41-s + 0.630·43-s + 0.496·45-s − 0.141·47-s − 0.744·51-s − 0.410·53-s + 2.93·55-s + 1.14·57-s − 0.864·59-s − 1.15·61-s + 2.07·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.328369992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.328369992\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 16.6T + 125T^{2} \) |
| 11 | \( 1 - 71.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 163.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 45.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 391.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 551.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 574.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 668.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 76.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 242.T + 9.12e5T^{2} \) |
| 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
−8.927954958875309952957026433376, −8.000520049231780603615460759143, −6.87555460371582779561365871624, −6.28880199858472558214832587842, −5.75576136128451806053761298531, −4.52515383051941230917677821815, −3.71907606166266782625378880684, −2.76073500848258481946540788565, −1.61247356156542201866858968265, −1.18177444876055460447460858930,
1.18177444876055460447460858930, 1.61247356156542201866858968265, 2.76073500848258481946540788565, 3.71907606166266782625378880684, 4.52515383051941230917677821815, 5.75576136128451806053761298531, 6.28880199858472558214832587842, 6.87555460371582779561365871624, 8.000520049231780603615460759143, 8.927954958875309952957026433376
