| L(s) = 1 | − 2-s − 1.73·3-s + 4-s − 0.267·5-s + 1.73·6-s + 7-s − 8-s + 0.267·10-s + 5.46·11-s − 1.73·12-s − 14-s + 0.464·15-s + 16-s − 3.46·17-s − 3.46·19-s − 0.267·20-s − 1.73·21-s − 5.46·22-s + 8.46·23-s + 1.73·24-s − 4.92·25-s + 5.19·27-s + 28-s + 8.92·29-s − 0.464·30-s − 0.535·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.00·3-s + 0.5·4-s − 0.119·5-s + 0.707·6-s + 0.377·7-s − 0.353·8-s + 0.0847·10-s + 1.64·11-s − 0.500·12-s − 0.267·14-s + 0.119·15-s + 0.250·16-s − 0.840·17-s − 0.794·19-s − 0.0599·20-s − 0.377·21-s − 1.16·22-s + 1.76·23-s + 0.353·24-s − 0.985·25-s + 1.00·27-s + 0.188·28-s + 1.65·29-s − 0.0847·30-s − 0.0962·31-s − 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(0.8619892291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8619892291\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 + 0.267T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 0.535T + 31T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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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
−8.821975471630970195759991037024, −8.501994808369764275341968301164, −7.31682761082116377290258510315, −6.49256881297276269871292154132, −6.25306406789705021879941006022, −5.02284102972145858711464536535, −4.33398205231086782078272777058, −3.11571271781561664056639438832, −1.77719600223170291185365882973, −0.71707200798988931466982101569,
0.71707200798988931466982101569, 1.77719600223170291185365882973, 3.11571271781561664056639438832, 4.33398205231086782078272777058, 5.02284102972145858711464536535, 6.25306406789705021879941006022, 6.49256881297276269871292154132, 7.31682761082116377290258510315, 8.501994808369764275341968301164, 8.821975471630970195759991037024
