| L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s + 6·11-s + 2·13-s − 2·14-s − 16-s + 2·17-s − 2·19-s − 20-s + 6·22-s − 2·23-s + 25-s + 2·26-s + 2·28-s + 29-s + 2·31-s + 5·32-s + 2·34-s − 2·35-s + 10·37-s − 2·38-s − 3·40-s − 2·41-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 + 1.80·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s + 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.185·29-s + 0.359·31-s + 0.883·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.324·38-s − 0.474·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.161933803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.161933803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−9.434356705021073856115254027826, −9.110855043614979188265787189711, −8.167158716049227693310970923226, −6.80875958973149432202684361265, −6.19042455071343211174006592245, −5.58777262856420392867756836576, −4.24191961212745454512819522376, −3.83216861154672491345510425928, −2.68651904520574431028594000101, −1.03452833414561398442010937116,
1.03452833414561398442010937116, 2.68651904520574431028594000101, 3.83216861154672491345510425928, 4.24191961212745454512819522376, 5.58777262856420392867756836576, 6.19042455071343211174006592245, 6.80875958973149432202684361265, 8.167158716049227693310970923226, 9.110855043614979188265787189711, 9.434356705021073856115254027826
