| L(s) = 1 | + 0.420·2-s − 3-s − 1.82·4-s + 1.49·5-s − 0.420·6-s − 1.60·8-s + 9-s + 0.630·10-s + 2.20·11-s + 1.82·12-s − 5.37·13-s − 1.49·15-s + 2.97·16-s + 3.93·17-s + 0.420·18-s − 4.05·19-s − 2.73·20-s + 0.925·22-s − 0.567·23-s + 1.60·24-s − 2.75·25-s − 2.25·26-s − 27-s + 3.22·29-s − 0.630·30-s + 7.94·31-s + 4.46·32-s + ⋯ |
| L(s) = 1 | + 0.297·2-s − 0.577·3-s − 0.911·4-s + 0.670·5-s − 0.171·6-s − 0.568·8-s + 0.333·9-s + 0.199·10-s + 0.663·11-s + 0.526·12-s − 1.48·13-s − 0.387·15-s + 0.742·16-s + 0.954·17-s + 0.0991·18-s − 0.929·19-s − 0.611·20-s + 0.197·22-s − 0.118·23-s + 0.328·24-s − 0.550·25-s − 0.443·26-s − 0.192·27-s + 0.598·29-s − 0.115·30-s + 1.42·31-s + 0.789·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.420T + 2T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 4.05T + 19T^{2} \) |
| 23 | \( 1 + 0.567T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 - 0.979T + 37T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 8.17T + 83T^{2} \) |
| 89 | \( 1 + 7.32T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.87600948294454984851828613704, −6.70997122002358903677865150980, −6.27566566795652208741467034492, −5.45218953501485965054639995639, −4.86164740977266422404060236141, −4.29788015023722817490649515160, −3.34061947117296819563839569248, −2.33419183311472278182007666850, −1.19973146259915185983708112430, 0,
1.19973146259915185983708112430, 2.33419183311472278182007666850, 3.34061947117296819563839569248, 4.29788015023722817490649515160, 4.86164740977266422404060236141, 5.45218953501485965054639995639, 6.27566566795652208741467034492, 6.70997122002358903677865150980, 7.87600948294454984851828613704
