| L(s) = 1 | − 2.94·3-s + 3.90·7-s + 5.67·9-s + 2.05·11-s − 4.35·13-s + 5.68·17-s − 4.55·19-s − 11.4·21-s + 4.68·23-s − 7.87·27-s + 6.49·29-s − 7.63·31-s − 6.06·33-s − 4.18·37-s + 12.8·39-s + 10.5·41-s − 7.38·43-s − 9.29·47-s + 8.22·49-s − 16.7·51-s − 3.50·53-s + 13.4·57-s + 2.15·59-s − 0.531·61-s + 22.1·63-s − 15.2·67-s − 13.7·69-s + ⋯ |
| L(s) = 1 | − 1.70·3-s + 1.47·7-s + 1.89·9-s + 0.620·11-s − 1.20·13-s + 1.37·17-s − 1.04·19-s − 2.50·21-s + 0.976·23-s − 1.51·27-s + 1.20·29-s − 1.37·31-s − 1.05·33-s − 0.688·37-s + 2.05·39-s + 1.64·41-s − 1.12·43-s − 1.35·47-s + 1.17·49-s − 2.34·51-s − 0.481·53-s + 1.77·57-s + 0.280·59-s − 0.0680·61-s + 2.78·63-s − 1.86·67-s − 1.66·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2.94T + 3T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 4.18T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 2.15T + 59T^{2} \) |
| 61 | \( 1 + 0.531T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 7.90T + 83T^{2} \) |
| 89 | \( 1 - 6.81T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.17365673284832624802088441350, −6.64874020618664229739365364220, −5.75143706653202704449935559494, −5.30913724869145596613419365140, −4.65929732111870120935402258283, −4.28554431613288996539859776994, −3.01249628903127421905859623390, −1.73463744994730343371409585791, −1.19137358184628197853000974460, 0,
1.19137358184628197853000974460, 1.73463744994730343371409585791, 3.01249628903127421905859623390, 4.28554431613288996539859776994, 4.65929732111870120935402258283, 5.30913724869145596613419365140, 5.75143706653202704449935559494, 6.64874020618664229739365364220, 7.17365673284832624802088441350
