| L(s) = 1 | − 3·3-s − 5·5-s − 30.3·7-s + 9·9-s − 20·11-s + 16.3·13-s + 15·15-s − 69.0·17-s − 86.3·19-s + 91.0·21-s + 34.3·23-s + 25·25-s − 27·27-s − 39.4·29-s + 217.·31-s + 60·33-s + 151.·35-s + 281.·37-s − 49.0·39-s + 342.·41-s − 373.·43-s − 45·45-s − 198.·47-s + 578.·49-s + 207.·51-s + 91.8·53-s + 100·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.63·7-s + 0.333·9-s − 0.548·11-s + 0.348·13-s + 0.258·15-s − 0.985·17-s − 1.04·19-s + 0.946·21-s + 0.311·23-s + 0.200·25-s − 0.192·27-s − 0.252·29-s + 1.25·31-s + 0.316·33-s + 0.732·35-s + 1.25·37-s − 0.201·39-s + 1.30·41-s − 1.32·43-s − 0.149·45-s − 0.616·47-s + 1.68·49-s + 0.568·51-s + 0.238·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6825453416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6825453416\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 30.3T + 343T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 69.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 34.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 373.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 91.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 49.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 309.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 651.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 850.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 964.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 724.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 433.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
| 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
−10.62117367932198304633167991235, −9.783540389538097253993076641118, −8.886413961600798197510415126819, −7.78542936854376283729280037132, −6.57089508151085956137660752948, −6.22863697435914677004708429154, −4.79904785431577203410673912992, −3.74360503870024231657469416728, −2.55249161658538461262675971936, −0.50573478296655405827117483282,
0.50573478296655405827117483282, 2.55249161658538461262675971936, 3.74360503870024231657469416728, 4.79904785431577203410673912992, 6.22863697435914677004708429154, 6.57089508151085956137660752948, 7.78542936854376283729280037132, 8.886413961600798197510415126819, 9.783540389538097253993076641118, 10.62117367932198304633167991235
