| L(s) = 1 | − 0.684·3-s − 2.03·5-s − 0.0739·7-s − 2.53·9-s − 6.59·11-s − 2.19·13-s + 1.38·15-s − 2.82·17-s + 3.57·19-s + 0.0505·21-s − 3.59·23-s − 0.870·25-s + 3.78·27-s − 0.888·29-s − 3.39·31-s + 4.51·33-s + 0.150·35-s − 1.90·37-s + 1.49·39-s − 5.01·41-s − 8.72·43-s + 5.14·45-s − 7.66·47-s − 6.99·49-s + 1.93·51-s − 7.04·53-s + 13.4·55-s + ⋯ |
| L(s) = 1 | − 0.394·3-s − 0.908·5-s − 0.0279·7-s − 0.844·9-s − 1.98·11-s − 0.607·13-s + 0.358·15-s − 0.684·17-s + 0.820·19-s + 0.0110·21-s − 0.750·23-s − 0.174·25-s + 0.728·27-s − 0.164·29-s − 0.609·31-s + 0.785·33-s + 0.0253·35-s − 0.312·37-s + 0.240·39-s − 0.783·41-s − 1.33·43-s + 0.767·45-s − 1.11·47-s − 0.999·49-s + 0.270·51-s − 0.967·53-s + 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04873569368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04873569368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 0.684T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 0.0739T + 7T^{2} \) |
| 11 | \( 1 + 6.59T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 0.888T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 0.987T + 61T^{2} \) |
| 67 | \( 1 + 0.134T + 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 0.111T + 83T^{2} \) |
| 89 | \( 1 + 0.114T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.147831298594353043931622936540, −7.47812424236533898059674913943, −6.74738573767432012509678226059, −5.79892903387767055635445160195, −5.14569922198280687998028327111, −4.68916129841040955523777265465, −3.50072506582552975749759898382, −2.90446084141600226655154658551, −1.94848559041024908230089954388, −0.10934809479731050087755952967,
0.10934809479731050087755952967, 1.94848559041024908230089954388, 2.90446084141600226655154658551, 3.50072506582552975749759898382, 4.68916129841040955523777265465, 5.14569922198280687998028327111, 5.79892903387767055635445160195, 6.74738573767432012509678226059, 7.47812424236533898059674913943, 8.147831298594353043931622936540
