L(s) = 1 | + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.733 − 0.680i)5-s + (0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (−1.21 + 0.825i)11-s + (−1.78 − 0.858i)13-s + (−0.988 − 0.149i)14-s + (0.955 − 0.294i)16-s + (0.5 − 0.866i)18-s + (−0.955 − 1.65i)19-s + (−0.623 + 0.781i)20-s + (0.914 + 1.14i)22-s + (0.365 − 0.930i)23-s + ⋯ |
L(s) = 1 | + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.733 − 0.680i)5-s + (0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (−1.21 + 0.825i)11-s + (−1.78 − 0.858i)13-s + (−0.988 − 0.149i)14-s + (0.955 − 0.294i)16-s + (0.5 − 0.866i)18-s + (−0.955 − 1.65i)19-s + (−0.623 + 0.781i)20-s + (0.914 + 1.14i)22-s + (0.365 − 0.930i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9302945357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9302945357\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.222 + 0.974i)T \) |
good | 3 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 11 | \( 1 + (1.21 - 0.825i)T + (0.365 - 0.930i)T^{2} \) |
| 13 | \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 19 | \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 0.246i)T + (0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.0546 + 0.728i)T + (-0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.326 - 0.302i)T + (0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-1.03 - 0.702i)T + (0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.353801140347946901669086238619, −8.279275714871793794354426318303, −7.62789553844312873759502747490, −6.89269375064304640615187079834, −5.27475694054981336244595850903, −4.74722201881592091939601758264, −4.39588267195403006113721212198, −2.59692237172061883867628336142, −2.20482955518857226378883974224, −0.67773379693508196874720765394,
1.86742996539146624611359309380, 2.94174481748259022142868262399, 4.20310790836306389819562610240, 5.18936910370915069712571946176, 5.83858398690941352794220477438, 6.47899717499539380318251236619, 7.40034715040015887322306368384, 7.946411449947095557487926544873, 8.952344967420845696294202021655, 9.701299975238874076434194286075