| L(s) = 1 | − 5.59·2-s + 23.2·4-s − 16.4·5-s − 10.5·7-s − 85.5·8-s + 92.2·10-s − 4.40·11-s − 64.7·13-s + 58.7·14-s + 292.·16-s − 7.65·17-s − 84.7·19-s − 384.·20-s + 24.6·22-s + 145.·23-s + 146.·25-s + 362.·26-s − 244.·28-s − 113.·29-s − 249.·31-s − 951.·32-s + 42.8·34-s + 173.·35-s + 95.9·37-s + 474.·38-s + 1.41e3·40-s + 391.·41-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.91·4-s − 1.47·5-s − 0.566·7-s − 3.78·8-s + 2.91·10-s − 0.120·11-s − 1.38·13-s + 1.12·14-s + 4.56·16-s − 0.109·17-s − 1.02·19-s − 4.29·20-s + 0.239·22-s + 1.32·23-s + 1.17·25-s + 2.73·26-s − 1.65·28-s − 0.724·29-s − 1.44·31-s − 5.25·32-s + 0.216·34-s + 0.836·35-s + 0.426·37-s + 2.02·38-s + 5.57·40-s + 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 5.59T + 8T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 7 | \( 1 + 10.5T + 343T^{2} \) |
| 11 | \( 1 + 4.40T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.65T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 113.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 95.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 200.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 392.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 614.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 104.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 705.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 813.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 985.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 932.T + 9.12e5T^{2} \) |
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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
−8.292738452967689085907169568328, −7.70097549709381761886424029864, −7.14106057687432334843371905636, −6.58853537673754609117883762100, −5.36035448690916917664640963206, −3.95542994992935254359444985930, −2.97395121432990734123193654250, −2.14111169129736582493343870832, −0.68446459393699690313856449089, 0,
0.68446459393699690313856449089, 2.14111169129736582493343870832, 2.97395121432990734123193654250, 3.95542994992935254359444985930, 5.36035448690916917664640963206, 6.58853537673754609117883762100, 7.14106057687432334843371905636, 7.70097549709381761886424029864, 8.292738452967689085907169568328
