| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 4·11-s + 14-s + 16-s − 6·17-s + 20-s − 4·22-s + 23-s + 25-s + 28-s + 8·29-s − 8·31-s + 32-s − 6·34-s + 35-s − 2·37-s + 40-s − 2·41-s + 8·43-s − 4·44-s + 46-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s − 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.188·28-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.312·41-s + 1.21·43-s − 0.603·44-s + 0.147·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.23109714774183, −15.68359725945777, −15.30822544375109, −14.68321948671278, −14.03852166703122, −13.57016442044792, −13.14072395888264, −12.53261672570506, −12.05383513860444, −11.17251939133715, −10.77791578244515, −10.40049601070697, −9.524801504102619, −8.864022741097774, −8.290689461797725, −7.508192376581483, −7.019854084010401, −6.224901698788214, −5.709182742074194, −4.889590982421879, −4.619610998640464, −3.674812016812130, −2.745382653794386, −2.297134313016430, −1.388821486213874, 0,
1.388821486213874, 2.297134313016430, 2.745382653794386, 3.674812016812130, 4.619610998640464, 4.889590982421879, 5.709182742074194, 6.224901698788214, 7.019854084010401, 7.508192376581483, 8.290689461797725, 8.864022741097774, 9.524801504102619, 10.40049601070697, 10.77791578244515, 11.17251939133715, 12.05383513860444, 12.53261672570506, 13.14072395888264, 13.57016442044792, 14.03852166703122, 14.68321948671278, 15.30822544375109, 15.68359725945777, 16.23109714774183
