| L(s) = 1 | − 1.73·2-s − 3-s + 0.999·4-s + 1.73·5-s + 1.73·6-s + 1.73·8-s − 2·9-s − 2.99·10-s + 5.19·11-s − 0.999·12-s − 1.73·15-s − 5·16-s − 6·17-s + 3.46·18-s + 1.73·19-s + 1.73·20-s − 9·22-s − 1.73·24-s − 2.00·25-s + 5·27-s + 3·29-s + 2.99·30-s + 1.73·31-s + 5.19·32-s − 5.19·33-s + 10.3·34-s − 1.99·36-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.499·4-s + 0.774·5-s + 0.707·6-s + 0.612·8-s − 0.666·9-s − 0.948·10-s + 1.56·11-s − 0.288·12-s − 0.447·15-s − 1.25·16-s − 1.45·17-s + 0.816·18-s + 0.397·19-s + 0.387·20-s − 1.91·22-s − 0.353·24-s − 0.400·25-s + 0.962·27-s + 0.557·29-s + 0.547·30-s + 0.311·31-s + 0.918·32-s − 0.904·33-s + 1.78·34-s − 0.333·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7705699653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7705699653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−8.089799109950804278884308989466, −6.98156067914161030349707848378, −6.52880596323780344614386008529, −6.04579796176578473624398674480, −5.02201739826478910840454363774, −4.45285153901390918758731254088, −3.40793554288504661636708410561, −2.22943641110778235167606980813, −1.53455364214477769241479963296, −0.56158128140756504732569902067,
0.56158128140756504732569902067, 1.53455364214477769241479963296, 2.22943641110778235167606980813, 3.40793554288504661636708410561, 4.45285153901390918758731254088, 5.02201739826478910840454363774, 6.04579796176578473624398674480, 6.52880596323780344614386008529, 6.98156067914161030349707848378, 8.089799109950804278884308989466
