| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 2·7-s − 2·9-s + 2·11-s + 2·12-s − 13-s + 4·14-s − 4·16-s + 2·17-s − 4·18-s + 2·21-s + 4·22-s − 9·23-s − 2·26-s − 5·27-s + 4·28-s + 5·29-s + 2·31-s − 8·32-s + 2·33-s + 4·34-s − 4·36-s − 8·37-s − 39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s − 16-s + 0.485·17-s − 0.942·18-s + 0.436·21-s + 0.852·22-s − 1.87·23-s − 0.392·26-s − 0.962·27-s + 0.755·28-s + 0.928·29-s + 0.359·31-s − 1.41·32-s + 0.348·33-s + 0.685·34-s − 2/3·36-s − 1.31·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.016363498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.016363498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.93421690815436842914513639852, −11.07632824989305080407651894750, −9.745339411961178288073885758254, −8.647969105414674323103722403718, −7.79278253980066871819257147589, −6.41493871731547586436271874024, −5.50737120413184810439847634712, −4.43338064598993885453060354142, −3.44740587880108288114904722723, −2.21415072325817682075527365861,
2.21415072325817682075527365861, 3.44740587880108288114904722723, 4.43338064598993885453060354142, 5.50737120413184810439847634712, 6.41493871731547586436271874024, 7.79278253980066871819257147589, 8.647969105414674323103722403718, 9.745339411961178288073885758254, 11.07632824989305080407651894750, 11.93421690815436842914513639852
