| L(s) = 1 | + 0.531·2-s − 1.71·4-s − 0.555·5-s + 1.54·7-s − 1.97·8-s − 0.295·10-s − 0.522·11-s − 4.41·13-s + 0.819·14-s + 2.38·16-s + 3.73·17-s + 19-s + 0.953·20-s − 0.277·22-s − 0.679·23-s − 4.69·25-s − 2.34·26-s − 2.64·28-s + 2.30·29-s + 11.0·31-s + 5.21·32-s + 1.98·34-s − 0.856·35-s − 9.40·37-s + 0.531·38-s + 1.09·40-s + 0.130·41-s + ⋯ |
| L(s) = 1 | + 0.375·2-s − 0.858·4-s − 0.248·5-s + 0.582·7-s − 0.698·8-s − 0.0933·10-s − 0.157·11-s − 1.22·13-s + 0.218·14-s + 0.596·16-s + 0.906·17-s + 0.229·19-s + 0.213·20-s − 0.0592·22-s − 0.141·23-s − 0.938·25-s − 0.460·26-s − 0.500·28-s + 0.428·29-s + 1.98·31-s + 0.922·32-s + 0.340·34-s − 0.144·35-s − 1.54·37-s + 0.0862·38-s + 0.173·40-s + 0.0204·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.531T + 2T^{2} \) |
| 5 | \( 1 + 0.555T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 + 0.522T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 23 | \( 1 + 0.679T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 0.130T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 2.13T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.60779466124172453716357814056, −6.82937132778585226279792131276, −5.88679539626354239044012340525, −5.18590384702586795275090032747, −4.77124280237355064340628494477, −4.00648226161971667901882604779, −3.23216175271062128500174493684, −2.37592837542751003954823201126, −1.15299303793360217711620368186, 0,
1.15299303793360217711620368186, 2.37592837542751003954823201126, 3.23216175271062128500174493684, 4.00648226161971667901882604779, 4.77124280237355064340628494477, 5.18590384702586795275090032747, 5.88679539626354239044012340525, 6.82937132778585226279792131276, 7.60779466124172453716357814056
