| L(s) = 1 | + 2-s − 2.75·3-s + 4-s + 2.81·5-s − 2.75·6-s + 5.04·7-s + 8-s + 4.60·9-s + 2.81·10-s + 4.03·11-s − 2.75·12-s − 5.05·13-s + 5.04·14-s − 7.74·15-s + 16-s + 0.263·17-s + 4.60·18-s − 0.722·19-s + 2.81·20-s − 13.9·21-s + 4.03·22-s − 4.15·23-s − 2.75·24-s + 2.89·25-s − 5.05·26-s − 4.42·27-s + 5.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.59·3-s + 0.5·4-s + 1.25·5-s − 1.12·6-s + 1.90·7-s + 0.353·8-s + 1.53·9-s + 0.888·10-s + 1.21·11-s − 0.796·12-s − 1.40·13-s + 1.34·14-s − 2.00·15-s + 0.250·16-s + 0.0639·17-s + 1.08·18-s − 0.165·19-s + 0.628·20-s − 3.03·21-s + 0.859·22-s − 0.866·23-s − 0.562·24-s + 0.579·25-s − 0.991·26-s − 0.851·27-s + 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.292142199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.292142199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 - 5.04T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 0.263T + 17T^{2} \) |
| 19 | \( 1 + 0.722T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 9.64T + 29T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 0.0540T + 43T^{2} \) |
| 47 | \( 1 + 0.360T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
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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
−7.79374198656890627457152599502, −7.12266088923904393390657527251, −6.30508077879606311078725520177, −5.89343616968943223791355968368, −5.18105706849712731251517163537, −4.66360297340618045051264089398, −4.22558823831612155667497855924, −2.51612593293121893780795337329, −1.74991441602258386411996532368, −1.02071419495720809769312659958,
1.02071419495720809769312659958, 1.74991441602258386411996532368, 2.51612593293121893780795337329, 4.22558823831612155667497855924, 4.66360297340618045051264089398, 5.18105706849712731251517163537, 5.89343616968943223791355968368, 6.30508077879606311078725520177, 7.12266088923904393390657527251, 7.79374198656890627457152599502
