| L(s) = 1 | + 2-s − 4-s + 5-s + 2·7-s − 3·8-s + 10-s + 11-s − 13-s + 2·14-s − 16-s − 2·17-s − 3·19-s − 20-s + 22-s + 25-s − 26-s − 2·28-s − 5·29-s − 31-s + 5·32-s − 2·34-s + 2·35-s − 5·37-s − 3·38-s − 3·40-s − 8·43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.316·10-s + 0.301·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.688·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.928·29-s − 0.179·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s − 0.821·37-s − 0.486·38-s − 0.474·40-s − 1.21·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−8.001744933811208998223633204229, −6.88998571418722905174168715319, −6.33278045796185517628127479444, −5.38996321225135854965253621754, −4.99464241156503102704353786204, −4.19193698414561509274460920140, −3.50099341228357834166105892802, −2.42766504319397140471243759738, −1.52809377903521356505656829590, 0,
1.52809377903521356505656829590, 2.42766504319397140471243759738, 3.50099341228357834166105892802, 4.19193698414561509274460920140, 4.99464241156503102704353786204, 5.38996321225135854965253621754, 6.33278045796185517628127479444, 6.88998571418722905174168715319, 8.001744933811208998223633204229
