| L(s) = 1 | − 2.59·2-s + 2.07·3-s + 4.73·4-s + 1.79·5-s − 5.39·6-s − 7.08·8-s + 1.31·9-s − 4.65·10-s + 6.32·11-s + 9.83·12-s + 5.48·13-s + 3.72·15-s + 8.92·16-s − 1.02·17-s − 3.42·18-s − 0.973·19-s + 8.48·20-s − 16.4·22-s + 3.75·23-s − 14.7·24-s − 1.78·25-s − 14.2·26-s − 3.49·27-s + 2.66·29-s − 9.67·30-s + 31-s − 8.99·32-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.19·3-s + 2.36·4-s + 0.802·5-s − 2.20·6-s − 2.50·8-s + 0.439·9-s − 1.47·10-s + 1.90·11-s + 2.83·12-s + 1.52·13-s + 0.962·15-s + 2.23·16-s − 0.248·17-s − 0.806·18-s − 0.223·19-s + 1.89·20-s − 3.49·22-s + 0.783·23-s − 3.00·24-s − 0.356·25-s − 2.79·26-s − 0.672·27-s + 0.495·29-s − 1.76·30-s + 0.179·31-s − 1.58·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533000620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533000620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 2.07T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 + 0.973T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 3.87T + 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 - 7.82T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 + 0.297T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.196092052479060781887752734351, −8.801166936984876868898638553524, −8.370088658437530214281974032715, −7.28663740999037501064046645191, −6.54155654463470059202720604645, −5.91597947117265986963088296678, −3.96493431047306963576901459810, −3.02243507748027733029788068912, −1.87990180109277651961270550345, −1.24354011141875429300477200323,
1.24354011141875429300477200323, 1.87990180109277651961270550345, 3.02243507748027733029788068912, 3.96493431047306963576901459810, 5.91597947117265986963088296678, 6.54155654463470059202720604645, 7.28663740999037501064046645191, 8.370088658437530214281974032715, 8.801166936984876868898638553524, 9.196092052479060781887752734351
