| L(s) = 1 | − 0.169·2-s + 6.82·3-s − 7.97·4-s − 1.80·5-s − 1.15·6-s + 16.1·7-s + 2.70·8-s + 19.5·9-s + 0.306·10-s − 11·11-s − 54.4·12-s − 2.73·14-s − 12.3·15-s + 63.3·16-s + 83.4·17-s − 3.31·18-s − 57.9·19-s + 14.4·20-s + 110.·21-s + 1.86·22-s − 51.4·23-s + 18.4·24-s − 121.·25-s − 50.5·27-s − 128.·28-s − 234.·29-s + 2.08·30-s + ⋯ |
| L(s) = 1 | − 0.0598·2-s + 1.31·3-s − 0.996·4-s − 0.161·5-s − 0.0786·6-s + 0.870·7-s + 0.119·8-s + 0.725·9-s + 0.00968·10-s − 0.301·11-s − 1.30·12-s − 0.0521·14-s − 0.212·15-s + 0.989·16-s + 1.19·17-s − 0.0434·18-s − 0.700·19-s + 0.161·20-s + 1.14·21-s + 0.0180·22-s − 0.466·23-s + 0.156·24-s − 0.973·25-s − 0.360·27-s − 0.867·28-s − 1.50·29-s + 0.0127·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.169T + 8T^{2} \) |
| 3 | \( 1 - 6.82T + 27T^{2} \) |
| 5 | \( 1 + 1.80T + 125T^{2} \) |
| 7 | \( 1 - 16.1T + 343T^{2} \) |
| 17 | \( 1 - 83.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 51.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 360.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 13.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 81.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 552.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 144.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 91.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 598.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 475.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 80.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 970.T + 9.12e5T^{2} \) |
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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
−8.395760754269282149700995789372, −7.87838048108722099801741456658, −7.48276613097598702877911549449, −5.85682304524365032509105008824, −5.16385716921538717008819099443, −4.03316129716209795312940175249, −3.64172501839712757543849405948, −2.39931528922821087021337440981, −1.45201933838114628643821310469, 0,
1.45201933838114628643821310469, 2.39931528922821087021337440981, 3.64172501839712757543849405948, 4.03316129716209795312940175249, 5.16385716921538717008819099443, 5.85682304524365032509105008824, 7.48276613097598702877911549449, 7.87838048108722099801741456658, 8.395760754269282149700995789372
