| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 4·11-s + 12-s − 2·13-s − 14-s + 16-s + 6·17-s − 18-s − 7·19-s + 21-s − 4·22-s + 4·23-s − 24-s − 5·25-s + 2·26-s + 27-s + 28-s + 6·29-s + 8·31-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.60·19-s + 0.218·21-s − 0.852·22-s + 0.834·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512824653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512824653\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.983446810027915574602765159065, −9.608544595039176707978133554463, −8.444478333767639654347771085439, −8.099828967754790268732602937394, −6.95685744317167035267505240379, −6.23118715573046571730255709124, −4.82232018667588426865161085748, −3.71578768433837293822165853769, −2.47904677229520964351007458459, −1.22017795057149232325880033056,
1.22017795057149232325880033056, 2.47904677229520964351007458459, 3.71578768433837293822165853769, 4.82232018667588426865161085748, 6.23118715573046571730255709124, 6.95685744317167035267505240379, 8.099828967754790268732602937394, 8.444478333767639654347771085439, 9.608544595039176707978133554463, 9.983446810027915574602765159065
