| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 27-s + 28-s − 29-s − 30-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 27-s + 28-s − 29-s − 30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5523663238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5523663238\) |
| \(L(1)\) |
\(\approx\) |
\(0.4962235224\) |
| \(L(1)\) |
\(\approx\) |
\(0.4962235224\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.02006103192075636813716029333, −20.56962474980431047713939653494, −19.57715233542028249122847790145, −18.66735453393764540063000179576, −18.18828891025682068265727989456, −17.628890496073410296812069898422, −16.45446422445576797607838002902, −16.26971007213998403332489276448, −15.31030376747591792419632826959, −14.68773565097826591900294342293, −13.25919582004840875586117742340, −12.02134608179228396301510991813, −11.84352532204787753085315009223, −10.99501205912433042824403039567, −10.358277857441798580563986953213, −9.52455434105758601241800050666, −8.14682892296090105038975938628, −7.79790755231885861509688885088, −7.12253461663248700384725885107, −5.885732231072962765538913606264, −5.18434051435326199927328179692, −4.13740227570593331636271033189, −2.9286320793893339243865637861, −1.617157987503675741180563954482, −0.66023922692598195778120740851,
0.66023922692598195778120740851, 1.617157987503675741180563954482, 2.9286320793893339243865637861, 4.13740227570593331636271033189, 5.18434051435326199927328179692, 5.885732231072962765538913606264, 7.12253461663248700384725885107, 7.79790755231885861509688885088, 8.14682892296090105038975938628, 9.52455434105758601241800050666, 10.358277857441798580563986953213, 10.99501205912433042824403039567, 11.84352532204787753085315009223, 12.02134608179228396301510991813, 13.25919582004840875586117742340, 14.68773565097826591900294342293, 15.31030376747591792419632826959, 16.26971007213998403332489276448, 16.45446422445576797607838002902, 17.628890496073410296812069898422, 18.18828891025682068265727989456, 18.66735453393764540063000179576, 19.57715233542028249122847790145, 20.56962474980431047713939653494, 21.02006103192075636813716029333
