| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 3·7-s + 8·10-s − 4·11-s + 6·14-s − 4·16-s − 17-s − 4·19-s − 8·20-s + 8·22-s − 6·23-s + 11·25-s − 6·28-s + 2·29-s + 7·31-s + 8·32-s + 2·34-s + 12·35-s − 2·37-s + 8·38-s − 2·41-s + 11·43-s − 8·44-s + 12·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 1.13·7-s + 2.52·10-s − 1.20·11-s + 1.60·14-s − 16-s − 0.242·17-s − 0.917·19-s − 1.78·20-s + 1.70·22-s − 1.25·23-s + 11/5·25-s − 1.13·28-s + 0.371·29-s + 1.25·31-s + 1.41·32-s + 0.342·34-s + 2.02·35-s − 0.328·37-s + 1.29·38-s − 0.312·41-s + 1.67·43-s − 1.20·44-s + 1.76·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−15.97982786629228, −15.60688582258288, −15.21654880682016, −14.39559766271876, −13.61974823697928, −13.01545602576671, −12.55713851015045, −11.93521355466865, −11.45715804734053, −10.80906999544535, −10.25558060362490, −10.07045347006894, −9.149130879341451, −8.650660135538694, −8.077665886825475, −7.845606391720963, −7.159300982849877, −6.661078442904420, −5.998055435563203, −4.930949424890872, −4.268596105996029, −3.767051091659232, −2.837258618350146, −2.344313890450583, −1.019930560911377, 0, 0,
1.019930560911377, 2.344313890450583, 2.837258618350146, 3.767051091659232, 4.268596105996029, 4.930949424890872, 5.998055435563203, 6.661078442904420, 7.159300982849877, 7.845606391720963, 8.077665886825475, 8.650660135538694, 9.149130879341451, 10.07045347006894, 10.25558060362490, 10.80906999544535, 11.45715804734053, 11.93521355466865, 12.55713851015045, 13.01545602576671, 13.61974823697928, 14.39559766271876, 15.21654880682016, 15.60688582258288, 15.97982786629228
