| L(s) = 1 | − 1.57i·2-s + 13.5·4-s − 15.5·5-s + 93.8i·7-s − 46.4i·8-s + 24.4i·10-s + (−60.9 + 104. i)11-s − 29.4i·13-s + 147.·14-s + 143.·16-s + 251. i·17-s + 80.2i·19-s − 210.·20-s + (164. + 95.9i)22-s + 702.·23-s + ⋯ |
| L(s) = 1 | − 0.393i·2-s + 0.845·4-s − 0.622·5-s + 1.91i·7-s − 0.726i·8-s + 0.244i·10-s + (−0.503 + 0.863i)11-s − 0.174i·13-s + 0.753·14-s + 0.559·16-s + 0.871i·17-s + 0.222i·19-s − 0.525·20-s + (0.340 + 0.198i)22-s + 1.32·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.42915 + 0.821148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42915 + 0.821148i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (60.9 - 104. i)T \) |
| good | 2 | \( 1 + 1.57iT - 16T^{2} \) |
| 5 | \( 1 + 15.5T + 625T^{2} \) |
| 7 | \( 1 - 93.8iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 29.4iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 251. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 80.2iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 702.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.44e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 115.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.07e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.88e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.59e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.19e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.89e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.77e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.25e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.59e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.75e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 6.74e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.61e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 9.33e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.10e4T + 8.85e7T^{2} \) |
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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
−12.75371182013711273348508053421, −12.29049403997542155215433071706, −11.40959083481008409010791538795, −10.30481593916785625574897778964, −8.976267436086318767520784447286, −7.82224652674453204409837049011, −6.47606022925196024453697113664, −5.17339145068908358757867581180, −3.17980221244277251547717425312, −1.95887254826252788516812995524,
0.75565528179752616384034221692, 3.10173910905737803332410122232, 4.59513577021001300303311903540, 6.37098559579856839547846992126, 7.40937151429648656868569351875, 8.065833661667666760351541943234, 9.956905600740331177697253009830, 11.05944032018945446674694133509, 11.57499542601646341734824925495, 13.27383075317722175102440049551
