| L(s) = 1 | + 2-s + 1.21·3-s + 4-s − 3.44·5-s + 1.21·6-s + 7-s + 8-s − 1.51·9-s − 3.44·10-s + 4.99·11-s + 1.21·12-s + 14-s − 4.18·15-s + 16-s − 7.06·17-s − 1.51·18-s + 4.99·19-s − 3.44·20-s + 1.21·21-s + 4.99·22-s + 5.66·23-s + 1.21·24-s + 6.83·25-s − 5.50·27-s + 28-s + 4.73·29-s − 4.18·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.702·3-s + 0.5·4-s − 1.53·5-s + 0.496·6-s + 0.377·7-s + 0.353·8-s − 0.506·9-s − 1.08·10-s + 1.50·11-s + 0.351·12-s + 0.267·14-s − 1.08·15-s + 0.250·16-s − 1.71·17-s − 0.357·18-s + 1.14·19-s − 0.769·20-s + 0.265·21-s + 1.06·22-s + 1.18·23-s + 0.248·24-s + 1.36·25-s − 1.05·27-s + 0.188·28-s + 0.878·29-s − 0.764·30-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
| \(L(1)\) |
\(\approx\) |
\(2.975282778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975282778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.21T + 3T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 - 4.99T + 11T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 6.00T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 3.79T + 53T^{2} \) |
| 59 | \( 1 - 0.563T + 59T^{2} \) |
| 61 | \( 1 + 9.60T + 61T^{2} \) |
| 67 | \( 1 + 7.50T + 67T^{2} \) |
| 71 | \( 1 + 4.20T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 - 0.0666T + 79T^{2} \) |
| 83 | \( 1 + 9.86T + 83T^{2} \) |
| 89 | \( 1 + 6.16T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.954431751108582978362766234235, −8.116311633126005267166781964203, −7.46511391426939960664839523051, −6.76953449987497319510947251537, −5.87022867530525538803934688385, −4.47725065280793356273113045612, −4.29667361490882676860891362632, −3.28271414299527705802169120105, −2.57431860602075574930642910856, −1.00611762990901930627528053617,
1.00611762990901930627528053617, 2.57431860602075574930642910856, 3.28271414299527705802169120105, 4.29667361490882676860891362632, 4.47725065280793356273113045612, 5.87022867530525538803934688385, 6.76953449987497319510947251537, 7.46511391426939960664839523051, 8.116311633126005267166781964203, 8.954431751108582978362766234235
