| L(s) = 1 | − 2-s + 2.62·3-s + 4-s − 3.76·5-s − 2.62·6-s − 8-s + 3.89·9-s + 3.76·10-s − 0.896·11-s + 2.62·12-s − 0.238·13-s − 9.87·15-s + 16-s + 2.38·17-s − 3.89·18-s − 3.25·19-s − 3.76·20-s + 0.896·22-s + 3.72·23-s − 2.62·24-s + 9.14·25-s + 0.238·26-s + 2.35·27-s + 29-s + 9.87·30-s − 5.49·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s − 1.68·5-s − 1.07·6-s − 0.353·8-s + 1.29·9-s + 1.18·10-s − 0.270·11-s + 0.758·12-s − 0.0661·13-s − 2.55·15-s + 0.250·16-s + 0.579·17-s − 0.918·18-s − 0.746·19-s − 0.841·20-s + 0.191·22-s + 0.777·23-s − 0.536·24-s + 1.82·25-s + 0.0467·26-s + 0.453·27-s + 0.185·29-s + 1.80·30-s − 0.986·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 11 | \( 1 + 0.896T + 11T^{2} \) |
| 13 | \( 1 + 0.238T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 + 3.25T + 19T^{2} \) |
| 23 | \( 1 - 3.72T + 23T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.52T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 + 4.98T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 9.87T + 79T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−8.427762034715509104996663938955, −7.85533210535536860650234663649, −7.32578528758779031940798322101, −6.60381908812622330265145343304, −5.10621194712055576512783339813, −4.10430862195103601437533742831, −3.38439670952891713185734576793, −2.78621417962213331583513707941, −1.54759693820661643498937660213, 0,
1.54759693820661643498937660213, 2.78621417962213331583513707941, 3.38439670952891713185734576793, 4.10430862195103601437533742831, 5.10621194712055576512783339813, 6.60381908812622330265145343304, 7.32578528758779031940798322101, 7.85533210535536860650234663649, 8.427762034715509104996663938955
