L(s) = 1 | − 1.99·3-s + 2.61·5-s + 4.58·7-s + 0.968·9-s + 4.22·11-s − 4.74·13-s − 5.21·15-s − 3.73·17-s − 3.23·19-s − 9.13·21-s + 6.76·23-s + 1.84·25-s + 4.04·27-s + 7.19·29-s + 1.31·31-s − 8.40·33-s + 11.9·35-s − 3.53·37-s + 9.45·39-s + 7.91·41-s − 2.99·43-s + 2.53·45-s + 4.75·47-s + 14.0·49-s + 7.44·51-s + 2.40·53-s + 11.0·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.16·5-s + 1.73·7-s + 0.322·9-s + 1.27·11-s − 1.31·13-s − 1.34·15-s − 0.906·17-s − 0.743·19-s − 1.99·21-s + 1.41·23-s + 0.368·25-s + 0.778·27-s + 1.33·29-s + 0.236·31-s − 1.46·33-s + 2.02·35-s − 0.580·37-s + 1.51·39-s + 1.23·41-s − 0.457·43-s + 0.377·45-s + 0.693·47-s + 2.00·49-s + 1.04·51-s + 0.330·53-s + 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.011175851\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.011175851\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 + 0.0714T + 67T^{2} \) |
| 71 | \( 1 - 3.10T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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
−8.763998169279552903597368577442, −7.53315030035024634820718286650, −6.78446900850313920567543496963, −6.20355958908122274623734542427, −5.40858836514045842396806100895, −4.77471052726112560881050514630, −4.37746889624423805927029796557, −2.62372931569775348681200798382, −1.82202397533453168977522798395, −0.912438228978432677554837170099,
0.912438228978432677554837170099, 1.82202397533453168977522798395, 2.62372931569775348681200798382, 4.37746889624423805927029796557, 4.77471052726112560881050514630, 5.40858836514045842396806100895, 6.20355958908122274623734542427, 6.78446900850313920567543496963, 7.53315030035024634820718286650, 8.763998169279552903597368577442