| L(s) = 1 | + 3.52i·2-s − 4.39·4-s + 9.17i·5-s + 12.6i·8-s − 32.2·10-s + 20.4i·11-s − 46.8·13-s − 79.8·16-s − 40.3i·20-s − 72.0·22-s + 40.8·25-s − 165. i·26-s − 179. i·32-s − 116.·40-s + 196. i·41-s + ⋯ |
| L(s) = 1 | + 1.24i·2-s − 0.549·4-s + 0.820i·5-s + 0.560i·8-s − 1.02·10-s + 0.561i·11-s − 1.00·13-s − 1.24·16-s − 0.450i·20-s − 0.698·22-s + 0.326·25-s − 1.24i·26-s − 0.991i·32-s − 0.460·40-s + 0.750i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.37343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37343i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 46.8T \) |
| good | 2 | \( 1 - 3.52iT - 8T^{2} \) |
| 5 | \( 1 - 9.17iT - 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 - 20.4iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 196. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 452T + 7.95e4T^{2} \) |
| 47 | \( 1 - 640. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 579. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 944.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.19e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 94.6iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.67e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−14.08284506730716299447225160560, −12.67498688710370025247363245093, −11.46190577922062520370950123984, −10.34929537658102652623724217554, −9.110946984540183029243930824246, −7.70088033977789600764664986507, −7.05361483036729596833337850764, −5.97098725603595227383763173865, −4.63880727924482051354889512484, −2.57712972834261045861582992102,
0.72804117484882406288662974146, 2.39050690219638523959115644465, 3.93244288367769326776970706608, 5.30028361417664673156297147422, 7.02740685829004620806370228726, 8.561013985526855336965885325755, 9.560862987777600341913144187226, 10.54108261706540084823297753016, 11.60160341664841070800638665387, 12.41310089328677827003160573590
