| L(s) = 1 | − 3.08i·3-s − 0.555i·7-s − 6.50·9-s + 4.65·11-s − 5.02i·13-s + 1.32i·17-s + 0.196·19-s − 1.71·21-s − i·23-s + 10.8i·27-s + 0.812·29-s − 2.11·31-s − 14.3i·33-s − 5.64i·37-s − 15.4·39-s + ⋯ |
| L(s) = 1 | − 1.78i·3-s − 0.209i·7-s − 2.16·9-s + 1.40·11-s − 1.39i·13-s + 0.322i·17-s + 0.0450·19-s − 0.373·21-s − 0.208i·23-s + 2.08i·27-s + 0.150·29-s − 0.379·31-s − 2.49i·33-s − 0.928i·37-s − 2.47·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458793017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458793017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 3.08iT - 3T^{2} \) |
| 7 | \( 1 + 0.555iT - 7T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 13 | \( 1 + 5.02iT - 13T^{2} \) |
| 17 | \( 1 - 1.32iT - 17T^{2} \) |
| 19 | \( 1 - 0.196T + 19T^{2} \) |
| 29 | \( 1 - 0.812T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 5.64iT - 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 1.66iT - 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 2.23iT - 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 - 6.91iT - 67T^{2} \) |
| 71 | \( 1 - 0.0120T + 71T^{2} \) |
| 73 | \( 1 + 15.2iT - 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 5.64iT - 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−7.69916961197109021041009148702, −7.23968630317741979924948044070, −6.53391672541393475795493889273, −5.95416898387189371132413670886, −5.23953550150774112962439739873, −3.93277425213611337285165369806, −3.12378295902883809445142020515, −2.11698395623133677032634983056, −1.28556164745002018135063204824, −0.41737490862760289317709817446,
1.50519136670070217635022270491, 2.79974815355376405602585664428, 3.64223857408394492642024258566, 4.30879632026447153912540167576, 4.74447618325584683929463090631, 5.72010371572184008887307984519, 6.40652709546045054265054612492, 7.19402786542196088827306978407, 8.391074547385065686818043891607, 8.986044574324703029379170836818
