L(s) = 1 | + 2-s + 2.20·3-s + 4-s + 0.344·5-s + 2.20·6-s + 1.98·7-s + 8-s + 1.84·9-s + 0.344·10-s + 1.66·11-s + 2.20·12-s + 1.87·13-s + 1.98·14-s + 0.757·15-s + 16-s − 5.72·17-s + 1.84·18-s + 2.63·19-s + 0.344·20-s + 4.37·21-s + 1.66·22-s + 5.59·23-s + 2.20·24-s − 4.88·25-s + 1.87·26-s − 2.54·27-s + 1.98·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.27·3-s + 0.5·4-s + 0.153·5-s + 0.898·6-s + 0.750·7-s + 0.353·8-s + 0.614·9-s + 0.108·10-s + 0.502·11-s + 0.635·12-s + 0.518·13-s + 0.530·14-s + 0.195·15-s + 0.250·16-s − 1.38·17-s + 0.434·18-s + 0.605·19-s + 0.0769·20-s + 0.953·21-s + 0.355·22-s + 1.16·23-s + 0.449·24-s − 0.976·25-s + 0.366·26-s − 0.489·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.642541970\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.642541970\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 - 0.344T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 - 2.63T + 19T^{2} \) |
| 23 | \( 1 - 5.59T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 0.338T + 71T^{2} \) |
| 73 | \( 1 + 8.10T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 + 5.22T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 + 5.50T + 97T^{2} \) |
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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
−8.496576089570680775180493291999, −7.67013506981468193422409668965, −7.14147469485557049592965053205, −6.16009483566324840576220406491, −5.43932520522996031953839431454, −4.35320100273794367639618563408, −3.95933536643963423518955431036, −2.89998428476845907129401217184, −2.26435357829424752119690656494, −1.30230903295269797043304796281,
1.30230903295269797043304796281, 2.26435357829424752119690656494, 2.89998428476845907129401217184, 3.95933536643963423518955431036, 4.35320100273794367639618563408, 5.43932520522996031953839431454, 6.16009483566324840576220406491, 7.14147469485557049592965053205, 7.67013506981468193422409668965, 8.496576089570680775180493291999