| L(s) = 1 | + (2.20 − 0.342i)5-s − 2.64·7-s − 3.00·11-s − 0.640i·13-s + 0.685·17-s − 5.28i·19-s − 2.27i·23-s + (4.76 − 1.51i)25-s − 8.15i·29-s − 2.96i·31-s + (−5.83 + 0.905i)35-s + 1.60i·37-s − 7.42i·41-s + 11.2·43-s − 4.19i·47-s + ⋯ |
| L(s) = 1 | + (0.988 − 0.153i)5-s − 0.997·7-s − 0.906·11-s − 0.177i·13-s + 0.166·17-s − 1.21i·19-s − 0.474i·23-s + (0.952 − 0.303i)25-s − 1.51i·29-s − 0.533i·31-s + (−0.986 + 0.152i)35-s + 0.264i·37-s − 1.15i·41-s + 1.71·43-s − 0.612i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0159 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0159 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315455290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315455290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.20 + 0.342i)T \) |
| good | 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 + 0.640iT - 13T^{2} \) |
| 17 | \( 1 - 0.685T + 17T^{2} \) |
| 19 | \( 1 + 5.28iT - 19T^{2} \) |
| 23 | \( 1 + 2.27iT - 23T^{2} \) |
| 29 | \( 1 + 8.15iT - 29T^{2} \) |
| 31 | \( 1 + 2.96iT - 31T^{2} \) |
| 37 | \( 1 - 1.60iT - 37T^{2} \) |
| 41 | \( 1 + 7.42iT - 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 4.19iT - 47T^{2} \) |
| 53 | \( 1 + 9.60T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 8.31iT - 97T^{2} \) |
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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
−9.447496716911909156939797347751, −8.687338762900585865040469651457, −7.68427553978920906365503198305, −6.76399074145628877891547741535, −6.01649543022676734988076961119, −5.30962374914599766536337950948, −4.27993393405154385512275214630, −2.93860666434224931245289488458, −2.27794092818240351758326529439, −0.51495800406504446697000280104,
1.48476823397720497847100331836, 2.72919986536143468058885618901, 3.48894162687608900191098239956, 4.86942938606285862081865277593, 5.73707685363915498110960589302, 6.34069755081366919786912253938, 7.23732818367385452814475057440, 8.118667190819643544167353874473, 9.192763018335257619496621047945, 9.678366695157717418672203353031
