| L(s) = 1 | + 4.96·2-s + 16.6·4-s + 1.79·5-s − 13.6·7-s + 42.9·8-s + 8.92·10-s − 52.4·11-s + 26.9·13-s − 67.9·14-s + 80.1·16-s + 58.4·17-s + 5.25·19-s + 29.9·20-s − 260.·22-s − 59.9·23-s − 121.·25-s + 133.·26-s − 227.·28-s − 257.·29-s − 120.·31-s + 54.0·32-s + 290.·34-s − 24.5·35-s + 88.4·37-s + 26.0·38-s + 77.2·40-s + 229.·41-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 2.08·4-s + 0.160·5-s − 0.739·7-s + 1.89·8-s + 0.282·10-s − 1.43·11-s + 0.575·13-s − 1.29·14-s + 1.25·16-s + 0.833·17-s + 0.0634·19-s + 0.334·20-s − 2.52·22-s − 0.543·23-s − 0.974·25-s + 1.00·26-s − 1.53·28-s − 1.65·29-s − 0.699·31-s + 0.298·32-s + 1.46·34-s − 0.118·35-s + 0.393·37-s + 0.111·38-s + 0.305·40-s + 0.875·41-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 | \( 1 - 4.96T + 8T^{2} \) |
| 5 | \( 1 - 1.79T + 125T^{2} \) |
| 7 | \( 1 + 13.6T + 343T^{2} \) |
| 11 | \( 1 + 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.25T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 88.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 286.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 71.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 104.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 201.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 799.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 521.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 989.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 85.7T + 9.12e5T^{2} \) |
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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
−7.87298519089613584740894247776, −7.47879152966630291509354930023, −6.33332150554489571136976564480, −5.77921094905166722374665627377, −5.27244444981018427107418254592, −4.19240623059413900435789250138, −3.46200114262218683828506895843, −2.73925929456985559674775892553, −1.77475749680853402876952105538, 0,
1.77475749680853402876952105538, 2.73925929456985559674775892553, 3.46200114262218683828506895843, 4.19240623059413900435789250138, 5.27244444981018427107418254592, 5.77921094905166722374665627377, 6.33332150554489571136976564480, 7.47879152966630291509354930023, 7.87298519089613584740894247776
