| L(s) = 1 | + 4.28·2-s − 13.6·4-s + 25·5-s + 202.·7-s − 195.·8-s + 107.·10-s + 121·11-s − 636.·13-s + 865.·14-s − 399.·16-s + 1.20e3·17-s + 1.15e3·19-s − 341.·20-s + 517.·22-s − 932.·23-s + 625·25-s − 2.72e3·26-s − 2.76e3·28-s − 1.80e3·29-s + 4.32e3·31-s + 4.54e3·32-s + 5.15e3·34-s + 5.05e3·35-s + 6.69e3·37-s + 4.93e3·38-s − 4.88e3·40-s − 1.35e4·41-s + ⋯ |
| L(s) = 1 | + 0.756·2-s − 0.427·4-s + 0.447·5-s + 1.55·7-s − 1.08·8-s + 0.338·10-s + 0.301·11-s − 1.04·13-s + 1.17·14-s − 0.389·16-s + 1.01·17-s + 0.733·19-s − 0.191·20-s + 0.228·22-s − 0.367·23-s + 0.200·25-s − 0.790·26-s − 0.666·28-s − 0.398·29-s + 0.808·31-s + 0.785·32-s + 0.764·34-s + 0.697·35-s + 0.803·37-s + 0.554·38-s − 0.483·40-s − 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.548753078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.548753078\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 4.28T + 32T^{2} \) |
| 7 | \( 1 - 202.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 636.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 932.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.69e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.48e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.22e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.25e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.07e4T + 8.58e9T^{2} \) |
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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
−10.04389199268413239203232623076, −9.357455517909056924501777981747, −8.262444926898538623640308073023, −7.53104759581805666595225697838, −6.13498862253515419978974995020, −5.14160289702265848219912393781, −4.72199308629403506167699583007, −3.47089865893418622142491418386, −2.16413733542718307022345682761, −0.888106748788288189372583808484,
0.888106748788288189372583808484, 2.16413733542718307022345682761, 3.47089865893418622142491418386, 4.72199308629403506167699583007, 5.14160289702265848219912393781, 6.13498862253515419978974995020, 7.53104759581805666595225697838, 8.262444926898538623640308073023, 9.357455517909056924501777981747, 10.04389199268413239203232623076
