| L(s) = 1 | − 0.889·2-s − 7.20·4-s − 6.34·5-s − 11.3·7-s + 13.5·8-s + 5.64·10-s − 54.9·11-s − 49.5·13-s + 10.0·14-s + 45.6·16-s + 9.76·17-s + 104.·19-s + 45.7·20-s + 48.9·22-s + 45.7·23-s − 84.7·25-s + 44.0·26-s + 81.6·28-s − 38.7·29-s + 159.·31-s − 148.·32-s − 8.68·34-s + 71.9·35-s + 152.·37-s − 92.6·38-s − 85.8·40-s + 270.·41-s + ⋯ |
| L(s) = 1 | − 0.314·2-s − 0.901·4-s − 0.567·5-s − 0.611·7-s + 0.597·8-s + 0.178·10-s − 1.50·11-s − 1.05·13-s + 0.192·14-s + 0.713·16-s + 0.139·17-s + 1.25·19-s + 0.511·20-s + 0.473·22-s + 0.414·23-s − 0.677·25-s + 0.332·26-s + 0.551·28-s − 0.248·29-s + 0.923·31-s − 0.822·32-s − 0.0438·34-s + 0.347·35-s + 0.676·37-s − 0.395·38-s − 0.339·40-s + 1.03·41-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 | \( 1 + 0.889T + 8T^{2} \) |
| 5 | \( 1 + 6.34T + 125T^{2} \) |
| 7 | \( 1 + 11.3T + 343T^{2} \) |
| 11 | \( 1 + 54.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.76T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 45.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 12.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 338.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 183.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 105.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 381.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 644.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 360.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 48.8T + 9.12e5T^{2} \) |
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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
−8.185497454965533237821661392617, −7.72281767655159703425560921399, −7.10632007388587653207700262325, −5.75088868094449199486418445220, −5.11542294367570151415182566635, −4.33483937395969666293700422760, −3.32276208689073016708881010259, −2.48394635305456238391260135082, −0.824631657490892840435592464286, 0,
0.824631657490892840435592464286, 2.48394635305456238391260135082, 3.32276208689073016708881010259, 4.33483937395969666293700422760, 5.11542294367570151415182566635, 5.75088868094449199486418445220, 7.10632007388587653207700262325, 7.72281767655159703425560921399, 8.185497454965533237821661392617
