| L(s) = 1 | − 30.9·2-s + 444.·4-s + 625·5-s + 2.40e3·7-s + 2.08e3·8-s − 1.93e4·10-s + 5.16e4·11-s − 1.29e5·13-s − 7.42e4·14-s − 2.92e5·16-s + 2.59e5·17-s + 9.03e4·19-s + 2.77e5·20-s − 1.59e6·22-s − 7.51e5·23-s + 3.90e5·25-s + 4.01e6·26-s + 1.06e6·28-s + 2.20e6·29-s + 3.10e6·31-s + 7.96e6·32-s − 8.02e6·34-s + 1.50e6·35-s − 1.42e7·37-s − 2.79e6·38-s + 1.30e6·40-s + 1.45e7·41-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.868·4-s + 0.447·5-s + 0.377·7-s + 0.180·8-s − 0.611·10-s + 1.06·11-s − 1.26·13-s − 0.516·14-s − 1.11·16-s + 0.753·17-s + 0.158·19-s + 0.388·20-s − 1.45·22-s − 0.560·23-s + 0.200·25-s + 1.72·26-s + 0.328·28-s + 0.578·29-s + 0.602·31-s + 1.34·32-s − 1.03·34-s + 0.169·35-s − 1.25·37-s − 0.217·38-s + 0.0805·40-s + 0.805·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.252278195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252278195\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 30.9T + 512T^{2} \) |
| 11 | \( 1 - 5.16e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.29e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.59e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.03e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.51e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.42e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.45e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.89e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.43e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.44e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.61e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.33e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.92e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.11e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.54e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.42e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.40e8T + 7.60e17T^{2} \) |
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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
−9.870987127299659834033128014257, −9.293202732492577720141946441712, −8.348259107127932925727677664374, −7.47171687350699000105467933958, −6.61961823223409538749833207148, −5.30570493406639795529939528095, −4.15432693126029565040611968431, −2.53119302362181091539746295291, −1.52403001673730540192208067005, −0.64393734655484929276314974562,
0.64393734655484929276314974562, 1.52403001673730540192208067005, 2.53119302362181091539746295291, 4.15432693126029565040611968431, 5.30570493406639795529939528095, 6.61961823223409538749833207148, 7.47171687350699000105467933958, 8.348259107127932925727677664374, 9.293202732492577720141946441712, 9.870987127299659834033128014257
