| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 7-s + 8-s + 9-s + 3·10-s + 4·11-s − 12-s − 14-s − 3·15-s + 16-s − 5·17-s + 18-s + 4·19-s + 3·20-s + 21-s + 4·22-s − 4·23-s − 24-s + 4·25-s − 27-s − 28-s − 9·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s − 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.917·19-s + 0.670·20-s + 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 4/5·25-s − 0.192·27-s − 0.188·28-s − 1.67·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.694736127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.694736127\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.59651909766934251262469557938, −6.99893235780422585459859260520, −6.24566484671825970342976792192, −5.87888360845478412386394270646, −5.35133223949148101771346080165, −4.25934840618155038188152704204, −3.85748862271742224104887070440, −2.57560771850699945901836711876, −1.95722367120046376464961573596, −0.931423654552130725482154305769,
0.931423654552130725482154305769, 1.95722367120046376464961573596, 2.57560771850699945901836711876, 3.85748862271742224104887070440, 4.25934840618155038188152704204, 5.35133223949148101771346080165, 5.87888360845478412386394270646, 6.24566484671825970342976792192, 6.99893235780422585459859260520, 7.59651909766934251262469557938
