| L(s) = 1 | − 2.72·2-s − 3-s + 5.40·4-s + 2.52·5-s + 2.72·6-s + 2.50·7-s − 9.27·8-s + 9-s − 6.87·10-s − 2.56·11-s − 5.40·12-s + 0.00898·13-s − 6.82·14-s − 2.52·15-s + 14.4·16-s − 3.23·17-s − 2.72·18-s + 3.79·19-s + 13.6·20-s − 2.50·21-s + 6.97·22-s − 6.52·23-s + 9.27·24-s + 1.37·25-s − 0.0244·26-s − 27-s + 13.5·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.70·4-s + 1.12·5-s + 1.11·6-s + 0.947·7-s − 3.27·8-s + 0.333·9-s − 2.17·10-s − 0.772·11-s − 1.56·12-s + 0.00249·13-s − 1.82·14-s − 0.651·15-s + 3.60·16-s − 0.784·17-s − 0.641·18-s + 0.869·19-s + 3.05·20-s − 0.546·21-s + 1.48·22-s − 1.36·23-s + 1.89·24-s + 0.275·25-s − 0.00479·26-s − 0.192·27-s + 2.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.00898T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 3.79T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 + 9.86T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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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
−7.66511898181115264442347874031, −6.95349131413464725230757426093, −6.34577151555082441604732838112, −5.53676526270325217542042962645, −5.15731574794672002414407372906, −3.70135777161593227431004967936, −2.31703260508351256445597000691, −2.03997629719779029385960594144, −1.15663274005987405125826410463, 0,
1.15663274005987405125826410463, 2.03997629719779029385960594144, 2.31703260508351256445597000691, 3.70135777161593227431004967936, 5.15731574794672002414407372906, 5.53676526270325217542042962645, 6.34577151555082441604732838112, 6.95349131413464725230757426093, 7.66511898181115264442347874031
