| L(s) = 1 | + 2-s + 1.96·3-s + 4-s + 2.90·5-s + 1.96·6-s + 7-s + 8-s + 0.876·9-s + 2.90·10-s − 2.70·11-s + 1.96·12-s + 14-s + 5.70·15-s + 16-s − 0.0298·17-s + 0.876·18-s + 1.70·19-s + 2.90·20-s + 1.96·21-s − 2.70·22-s + 7.87·23-s + 1.96·24-s + 3.41·25-s − 4.18·27-s + 28-s − 6.57·29-s + 5.70·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.13·3-s + 0.5·4-s + 1.29·5-s + 0.803·6-s + 0.377·7-s + 0.353·8-s + 0.292·9-s + 0.917·10-s − 0.814·11-s + 0.568·12-s + 0.267·14-s + 1.47·15-s + 0.250·16-s − 0.00722·17-s + 0.206·18-s + 0.391·19-s + 0.648·20-s + 0.429·21-s − 0.576·22-s + 1.64·23-s + 0.401·24-s + 0.682·25-s − 0.804·27-s + 0.188·28-s − 1.22·29-s + 1.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.246219722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.246219722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 17 | \( 1 + 0.0298T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 7.87T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 8.36T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 9.31T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 9.28T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.0106T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−9.121289909730673516895145623902, −8.168397315252369674305320193553, −7.47840420016499634435995802311, −6.63760928175725316647156892575, −5.53107477568738440208897828348, −5.24943507538900248250541211530, −4.03615688238680697737233922979, −2.96490331915325237402907049284, −2.45215778537481662176220641522, −1.50358503386776561157536397168,
1.50358503386776561157536397168, 2.45215778537481662176220641522, 2.96490331915325237402907049284, 4.03615688238680697737233922979, 5.24943507538900248250541211530, 5.53107477568738440208897828348, 6.63760928175725316647156892575, 7.47840420016499634435995802311, 8.168397315252369674305320193553, 9.121289909730673516895145623902
