| L(s) = 1 | + 2-s − 1.57·3-s + 4-s + 1.25·5-s − 1.57·6-s − 0.329·7-s + 8-s − 0.504·9-s + 1.25·10-s + 2.98·11-s − 1.57·12-s − 0.329·14-s − 1.98·15-s + 16-s − 5.34·17-s − 0.504·18-s − 19-s + 1.25·20-s + 0.520·21-s + 2.98·22-s + 2.43·23-s − 1.57·24-s − 3.41·25-s + 5.53·27-s − 0.329·28-s − 7.29·29-s − 1.98·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.912·3-s + 0.5·4-s + 0.562·5-s − 0.644·6-s − 0.124·7-s + 0.353·8-s − 0.168·9-s + 0.397·10-s + 0.899·11-s − 0.456·12-s − 0.0880·14-s − 0.513·15-s + 0.250·16-s − 1.29·17-s − 0.118·18-s − 0.229·19-s + 0.281·20-s + 0.113·21-s + 0.636·22-s + 0.508·23-s − 0.322·24-s − 0.683·25-s + 1.06·27-s − 0.0622·28-s − 1.35·29-s − 0.362·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 0.329T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 0.248T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 0.687T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 8.51T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−7.42019191717234483362179562163, −6.54919562720631012756388605772, −6.22971158008505633192616140727, −5.64645913709735284777265007115, −4.81529071626626381512027330496, −4.23142696921856898883690192164, −3.28846560796740865708187962007, −2.31084601965391748551679355168, −1.42451458459522832245766028310, 0,
1.42451458459522832245766028310, 2.31084601965391748551679355168, 3.28846560796740865708187962007, 4.23142696921856898883690192164, 4.81529071626626381512027330496, 5.64645913709735284777265007115, 6.22971158008505633192616140727, 6.54919562720631012756388605772, 7.42019191717234483362179562163
