| L(s) = 1 | + 2.74·2-s − 1.51·3-s + 5.50·4-s − 0.684·5-s − 4.14·6-s − 1.23·7-s + 9.61·8-s − 0.708·9-s − 1.87·10-s + 4.93·11-s − 8.33·12-s + 4.27·13-s − 3.38·14-s + 1.03·15-s + 15.3·16-s − 1.08·17-s − 1.94·18-s + 1.43·19-s − 3.77·20-s + 1.87·21-s + 13.5·22-s + 3.12·23-s − 14.5·24-s − 4.53·25-s + 11.7·26-s + 5.61·27-s − 6.80·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.874·3-s + 2.75·4-s − 0.306·5-s − 1.69·6-s − 0.466·7-s + 3.39·8-s − 0.236·9-s − 0.593·10-s + 1.48·11-s − 2.40·12-s + 1.18·13-s − 0.904·14-s + 0.267·15-s + 3.83·16-s − 0.262·17-s − 0.457·18-s + 0.330·19-s − 0.843·20-s + 0.408·21-s + 2.88·22-s + 0.652·23-s − 2.97·24-s − 0.906·25-s + 2.29·26-s + 1.08·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.037620450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.037620450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 + 0.684T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 1.99T + 71T^{2} \) |
| 73 | \( 1 + 7.78T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 7.41T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 + 8.04T + 97T^{2} \) |
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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
−10.56881252364901450130057891216, −9.281049762009056803376979735002, −7.992287920716067776202411774453, −6.71390374035284717537104789196, −6.41351268311492329550147150057, −5.67267946249969709850244006326, −4.72738512972729862167166334481, −3.81705324000764174666019802809, −3.13905572779118192491547987485, −1.48106278673073476220837010933,
1.48106278673073476220837010933, 3.13905572779118192491547987485, 3.81705324000764174666019802809, 4.72738512972729862167166334481, 5.67267946249969709850244006326, 6.41351268311492329550147150057, 6.71390374035284717537104789196, 7.992287920716067776202411774453, 9.281049762009056803376979735002, 10.56881252364901450130057891216
