| L(s) = 1 | − 2-s + 4-s + 2.89·5-s + 0.647·7-s − 8-s − 2.89·10-s − 3.72·11-s + 0.715·13-s − 0.647·14-s + 16-s − 0.587·17-s + 6.18·19-s + 2.89·20-s + 3.72·22-s − 7.30·23-s + 3.37·25-s − 0.715·26-s + 0.647·28-s − 0.536·29-s − 8.61·31-s − 32-s + 0.587·34-s + 1.87·35-s − 2.58·37-s − 6.18·38-s − 2.89·40-s + 6.06·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.29·5-s + 0.244·7-s − 0.353·8-s − 0.915·10-s − 1.12·11-s + 0.198·13-s − 0.172·14-s + 0.250·16-s − 0.142·17-s + 1.41·19-s + 0.647·20-s + 0.794·22-s − 1.52·23-s + 0.675·25-s − 0.140·26-s + 0.122·28-s − 0.0995·29-s − 1.54·31-s − 0.176·32-s + 0.100·34-s + 0.316·35-s − 0.424·37-s − 1.00·38-s − 0.457·40-s + 0.946·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 - 0.647T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 0.715T + 13T^{2} \) |
| 17 | \( 1 + 0.587T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 + 7.30T + 23T^{2} \) |
| 29 | \( 1 + 0.536T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 6.06T + 41T^{2} \) |
| 43 | \( 1 - 0.0254T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.72669655052611214960007725586, −6.84223006280121267293601923309, −6.04656143293067172014546479906, −5.52812503848959318166387132232, −4.97251828390379902575885692175, −3.73721621452997557185021955256, −2.81842927078067047953353323981, −2.04793349931647712090571763998, −1.40610189709168655062618805534, 0,
1.40610189709168655062618805534, 2.04793349931647712090571763998, 2.81842927078067047953353323981, 3.73721621452997557185021955256, 4.97251828390379902575885692175, 5.52812503848959318166387132232, 6.04656143293067172014546479906, 6.84223006280121267293601923309, 7.72669655052611214960007725586
