| L(s) = 1 | − 1.11·2-s + 3.26·3-s − 0.751·4-s + 5-s − 3.64·6-s − 3.89·7-s + 3.07·8-s + 7.65·9-s − 1.11·10-s + 2.23·11-s − 2.45·12-s + 0.399·13-s + 4.35·14-s + 3.26·15-s − 1.93·16-s + 1.78·17-s − 8.54·18-s + 0.862·19-s − 0.751·20-s − 12.7·21-s − 2.49·22-s + 7.16·23-s + 10.0·24-s + 25-s − 0.446·26-s + 15.1·27-s + 2.93·28-s + ⋯ |
| L(s) = 1 | − 0.789·2-s + 1.88·3-s − 0.375·4-s + 0.447·5-s − 1.48·6-s − 1.47·7-s + 1.08·8-s + 2.55·9-s − 0.353·10-s + 0.672·11-s − 0.708·12-s + 0.110·13-s + 1.16·14-s + 0.842·15-s − 0.482·16-s + 0.431·17-s − 2.01·18-s + 0.197·19-s − 0.168·20-s − 2.77·21-s − 0.531·22-s + 1.49·23-s + 2.04·24-s + 0.200·25-s − 0.0876·26-s + 2.92·27-s + 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936719771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936719771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 0.399T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 0.862T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 0.167T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 + 9.64T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
| 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
−9.367332892617844222837643902385, −9.103390361638132603196693324855, −8.521558672178285133983456808169, −7.21354094431284361033443476303, −7.12459568414576535328208119394, −5.55992318608486370903672832585, −4.07101520034687968836617630160, −3.47896859282886527638559648327, −2.44986839088822207759792673097, −1.17882501799111267409416074103,
1.17882501799111267409416074103, 2.44986839088822207759792673097, 3.47896859282886527638559648327, 4.07101520034687968836617630160, 5.55992318608486370903672832585, 7.12459568414576535328208119394, 7.21354094431284361033443476303, 8.521558672178285133983456808169, 9.103390361638132603196693324855, 9.367332892617844222837643902385
