| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 12-s + 2·13-s − 2·14-s + 16-s + 2·17-s − 18-s − 2·21-s + 2·23-s + 24-s − 2·26-s − 27-s + 2·28-s + 4·29-s + 4·31-s − 32-s − 2·34-s + 36-s + 2·37-s − 2·39-s − 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.436·21-s + 0.417·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.320·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.79914571710953, −14.27173989479665, −13.53317321213823, −13.26972530955898, −12.36664981963929, −12.07332490966453, −11.45615226202501, −11.17053394265580, −10.53292019987963, −10.12128391721421, −9.616198070145595, −8.907490419155043, −8.402285409837771, −7.947510694062808, −7.466757246403761, −6.584120100940754, −6.460676335111329, −5.650064243498611, −4.986363691241046, −4.648354119454114, −3.696696782018017, −3.119590678479340, −2.262436474312720, −1.458515400690080, −1.023325517761366, 0,
1.023325517761366, 1.458515400690080, 2.262436474312720, 3.119590678479340, 3.696696782018017, 4.648354119454114, 4.986363691241046, 5.650064243498611, 6.460676335111329, 6.584120100940754, 7.466757246403761, 7.947510694062808, 8.402285409837771, 8.907490419155043, 9.616198070145595, 10.12128391721421, 10.53292019987963, 11.17053394265580, 11.45615226202501, 12.07332490966453, 12.36664981963929, 13.26972530955898, 13.53317321213823, 14.27173989479665, 14.79914571710953
