| L(s) = 1 | + 1.44·2-s + 0.0382·3-s + 0.100·4-s − 3.85·5-s + 0.0554·6-s + 1.66·7-s − 2.75·8-s − 2.99·9-s − 5.59·10-s + 1.88·11-s + 0.00384·12-s + 2.40·14-s − 0.147·15-s − 4.19·16-s − 3.75·17-s − 4.34·18-s − 3.26·19-s − 0.388·20-s + 0.0636·21-s + 2.72·22-s − 5.31·23-s − 0.105·24-s + 9.89·25-s − 0.229·27-s + 0.167·28-s + 3.11·29-s − 0.214·30-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0220·3-s + 0.0503·4-s − 1.72·5-s + 0.0226·6-s + 0.628·7-s − 0.973·8-s − 0.999·9-s − 1.76·10-s + 0.566·11-s + 0.00111·12-s + 0.643·14-s − 0.0381·15-s − 1.04·16-s − 0.909·17-s − 1.02·18-s − 0.750·19-s − 0.0868·20-s + 0.0138·21-s + 0.581·22-s − 1.10·23-s − 0.0214·24-s + 1.97·25-s − 0.0441·27-s + 0.0316·28-s + 0.577·29-s − 0.0390·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135642902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135642902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 0.0382T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 - 6.11T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 + 0.735T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
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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
−8.212736272085001545702301777573, −7.53046648903365700054036568453, −6.62133038642940336573834042193, −5.95135150928396680482007641256, −5.05097532386113247860130824398, −4.32519710232892844018729750355, −3.96927056394785118356568109712, −3.18529965950838394347766605893, −2.22685859685820700971705112905, −0.46962346214478409771466926672,
0.46962346214478409771466926672, 2.22685859685820700971705112905, 3.18529965950838394347766605893, 3.96927056394785118356568109712, 4.32519710232892844018729750355, 5.05097532386113247860130824398, 5.95135150928396680482007641256, 6.62133038642940336573834042193, 7.53046648903365700054036568453, 8.212736272085001545702301777573
