| L(s) = 1 | + 0.634·3-s + 1.51·5-s − 2.59·9-s − 1.78·11-s + 13-s + 0.962·15-s + 4.11·17-s − 4.28·19-s + 0.880·23-s − 2.70·25-s − 3.55·27-s − 8.23·29-s + 3.23·31-s − 1.13·33-s − 1.86·37-s + 0.634·39-s − 3.55·41-s + 4.47·43-s − 3.93·45-s − 0.457·47-s + 2.61·51-s − 5.25·53-s − 2.70·55-s − 2.72·57-s + 14.1·59-s − 11.8·61-s + 1.51·65-s + ⋯ |
| L(s) = 1 | + 0.366·3-s + 0.677·5-s − 0.865·9-s − 0.538·11-s + 0.277·13-s + 0.248·15-s + 0.997·17-s − 0.984·19-s + 0.183·23-s − 0.540·25-s − 0.683·27-s − 1.52·29-s + 0.580·31-s − 0.197·33-s − 0.305·37-s + 0.101·39-s − 0.554·41-s + 0.682·43-s − 0.586·45-s − 0.0667·47-s + 0.365·51-s − 0.721·53-s − 0.364·55-s − 0.360·57-s + 1.83·59-s − 1.51·61-s + 0.187·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.634T + 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 - 0.880T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 + 3.55T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 0.457T + 47T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 + 9.91T + 71T^{2} \) |
| 73 | \( 1 + 9.03T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 1.33T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−7.911803222320226413620911925095, −7.29925314452933148244980665011, −6.19256153717064378686556904351, −5.78705450666196783394445855708, −5.09222576161665211158359368370, −4.02491689213859476093064814224, −3.18631020649018409305642824669, −2.41514033585062421850538704392, −1.55124893552079090343972975559, 0,
1.55124893552079090343972975559, 2.41514033585062421850538704392, 3.18631020649018409305642824669, 4.02491689213859476093064814224, 5.09222576161665211158359368370, 5.78705450666196783394445855708, 6.19256153717064378686556904351, 7.29925314452933148244980665011, 7.911803222320226413620911925095
