| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s + 3·7-s − 2·9-s − 2·10-s − 3·11-s − 2·12-s + 4·13-s − 6·14-s − 15-s − 4·16-s − 2·17-s + 4·18-s + 2·20-s − 3·21-s + 6·22-s + 4·23-s + 25-s − 8·26-s + 5·27-s + 6·28-s + 2·30-s + 7·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s − 2/3·9-s − 0.632·10-s − 0.904·11-s − 0.577·12-s + 1.10·13-s − 1.60·14-s − 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s + 0.447·20-s − 0.654·21-s + 1.27·22-s + 0.834·23-s + 1/5·25-s − 1.56·26-s + 0.962·27-s + 1.13·28-s + 0.365·30-s + 1.25·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6249954625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6249954625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 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
−11.03584593329883704099928821271, −10.52616204381071415319152482689, −9.392092997878299416453084703285, −8.455377116229754512883998757977, −8.001604417681190038866635973448, −6.73539441646475021862154518181, −5.64384609470310284471770398546, −4.62450127466833198633428372182, −2.48321106254252969564317387787, −1.01023555705729953744464184528,
1.01023555705729953744464184528, 2.48321106254252969564317387787, 4.62450127466833198633428372182, 5.64384609470310284471770398546, 6.73539441646475021862154518181, 8.001604417681190038866635973448, 8.455377116229754512883998757977, 9.392092997878299416453084703285, 10.52616204381071415319152482689, 11.03584593329883704099928821271
