| L(s) = 1 | + 4·2-s + 16·4-s + 41.6·5-s − 203.·7-s + 64·8-s + 166.·10-s + 470.·11-s + 483.·13-s − 813.·14-s + 256·16-s + 1.25e3·17-s + 1.97e3·19-s + 665.·20-s + 1.88e3·22-s + 478.·23-s − 1.39e3·25-s + 1.93e3·26-s − 3.25e3·28-s − 1.16e3·29-s − 2.37e3·31-s + 1.02e3·32-s + 5.03e3·34-s − 8.45e3·35-s + 8.18e3·37-s + 7.91e3·38-s + 2.66e3·40-s + 1.75e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.744·5-s − 1.56·7-s + 0.353·8-s + 0.526·10-s + 1.17·11-s + 0.792·13-s − 1.10·14-s + 0.250·16-s + 1.05·17-s + 1.25·19-s + 0.372·20-s + 0.828·22-s + 0.188·23-s − 0.446·25-s + 0.560·26-s − 0.784·28-s − 0.256·29-s − 0.443·31-s + 0.176·32-s + 0.747·34-s − 1.16·35-s + 0.982·37-s + 0.889·38-s + 0.263·40-s + 1.62·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.395494516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.395494516\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 41.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 203.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 470.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 483.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.97e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 478.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.16e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.45e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.03e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.04e4T + 8.58e9T^{2} \) |
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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
−12.21600822342785202965578866417, −11.09576128567755082825236773013, −9.744633575377046028011575150996, −9.305572739765755118455182466684, −7.46634189619773889852536877796, −6.25905612752191172459902515154, −5.75277227162910148815057501586, −3.92369645003305813265934001499, −2.96445611308856911671548862889, −1.18197216325106220160834010274,
1.18197216325106220160834010274, 2.96445611308856911671548862889, 3.92369645003305813265934001499, 5.75277227162910148815057501586, 6.25905612752191172459902515154, 7.46634189619773889852536877796, 9.305572739765755118455182466684, 9.744633575377046028011575150996, 11.09576128567755082825236773013, 12.21600822342785202965578866417
