| L(s) = 1 | − 3.17·3-s − 5-s + 0.630·7-s + 7.04·9-s − 5.70·11-s − 1.80·13-s + 3.17·15-s + 2·17-s + 19-s − 2·21-s − 0.630·23-s + 25-s − 12.8·27-s + 10.4·29-s − 7.75·31-s + 18.0·33-s − 0.630·35-s − 3.80·37-s + 5.70·39-s − 4.49·41-s + 9.80·43-s − 7.04·45-s + 0.630·47-s − 6.60·49-s − 6.34·51-s + 7.80·53-s + 5.70·55-s + ⋯ |
| L(s) = 1 | − 1.83·3-s − 0.447·5-s + 0.238·7-s + 2.34·9-s − 1.72·11-s − 0.499·13-s + 0.818·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s − 0.131·23-s + 0.200·25-s − 2.47·27-s + 1.94·29-s − 1.39·31-s + 3.15·33-s − 0.106·35-s − 0.624·37-s + 0.914·39-s − 0.702·41-s + 1.49·43-s − 1.05·45-s + 0.0920·47-s − 0.943·49-s − 0.887·51-s + 1.07·53-s + 0.769·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 0.630T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 - 0.630T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 9.66T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.57341621249441350487245104691, −6.99819329057257606616194639788, −6.17951619778166335729804448237, −5.41268512723719867594670927371, −5.03696367395824352363636136524, −4.41937508413274422834326426203, −3.31466219632723102692444008179, −2.18127591706728015314077057080, −0.907692605326591438652290872169, 0,
0.907692605326591438652290872169, 2.18127591706728015314077057080, 3.31466219632723102692444008179, 4.41937508413274422834326426203, 5.03696367395824352363636136524, 5.41268512723719867594670927371, 6.17951619778166335729804448237, 6.99819329057257606616194639788, 7.57341621249441350487245104691
