| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s − 2·14-s − 16-s + 17-s − 2·19-s − 20-s + 25-s − 2·28-s + 10·29-s + 4·31-s − 5·32-s − 34-s + 2·35-s − 2·37-s + 2·38-s + 3·40-s + 6·41-s + 6·43-s − 8·47-s − 3·49-s − 50-s + 2·53-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 − 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.377·28-s + 1.85·29-s + 0.718·31-s − 0.883·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s + 0.324·38-s + 0.474·40-s + 0.937·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112306634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112306634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.51389174328912, −12.84679397367599, −12.82031974787664, −11.86991846739905, −11.58450974193926, −10.92633113131193, −10.44139568115698, −10.04386534020807, −9.718997045989248, −8.922134407087061, −8.731107451317478, −8.065631189997140, −7.905400792369218, −7.132573491465278, −6.596899065583165, −6.076771325490199, −5.251027509803506, −5.026797503916247, −4.333444925768322, −3.976113274813420, −3.046857113239134, −2.416140058588596, −1.760281403369483, −1.081012873657020, −0.5857149732297745,
0.5857149732297745, 1.081012873657020, 1.760281403369483, 2.416140058588596, 3.046857113239134, 3.976113274813420, 4.333444925768322, 5.026797503916247, 5.251027509803506, 6.076771325490199, 6.596899065583165, 7.132573491465278, 7.905400792369218, 8.065631189997140, 8.731107451317478, 8.922134407087061, 9.718997045989248, 10.04386534020807, 10.44139568115698, 10.92633113131193, 11.58450974193926, 11.86991846739905, 12.82031974787664, 12.84679397367599, 13.51389174328912
