| L(s) = 1 | − 1.70·2-s − 2.85·3-s + 0.921·4-s + 3.70·5-s + 4.87·6-s − 1.37·7-s + 1.84·8-s + 5.12·9-s − 6.33·10-s + 3.92·11-s − 2.62·12-s + 3.78·13-s + 2.34·14-s − 10.5·15-s − 4.99·16-s + 5.66·17-s − 8.75·18-s + 0.487·19-s + 3.41·20-s + 3.91·21-s − 6.71·22-s + 4.17·23-s − 5.25·24-s + 8.74·25-s − 6.47·26-s − 6.05·27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 1.64·3-s + 0.460·4-s + 1.65·5-s + 1.98·6-s − 0.518·7-s + 0.651·8-s + 1.70·9-s − 2.00·10-s + 1.18·11-s − 0.758·12-s + 1.05·13-s + 0.627·14-s − 2.72·15-s − 1.24·16-s + 1.37·17-s − 2.06·18-s + 0.111·19-s + 0.763·20-s + 0.853·21-s − 1.43·22-s + 0.871·23-s − 1.07·24-s + 1.74·25-s − 1.27·26-s − 1.16·27-s − 0.239·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7105461370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7105461370\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 0.487T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 7.29T + 97T^{2} \) |
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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
−10.07216800509709723827484188164, −9.292392976483061011782234138390, −8.785482070441260101647195739845, −7.16398870733952680802211470745, −6.65623442192612723017569268626, −5.71274346068698655529108924502, −5.30521435993619391085989366095, −3.76092635425799587779550149193, −1.68626792281569980646711754544, −0.957223885435891167557797151961,
0.957223885435891167557797151961, 1.68626792281569980646711754544, 3.76092635425799587779550149193, 5.30521435993619391085989366095, 5.71274346068698655529108924502, 6.65623442192612723017569268626, 7.16398870733952680802211470745, 8.785482070441260101647195739845, 9.292392976483061011782234138390, 10.07216800509709723827484188164
