L(s) = 1 | + 2-s − 0.857·3-s + 4-s − 2.26·5-s − 0.857·6-s − 3.52·7-s + 8-s − 2.26·9-s − 2.26·10-s − 1.16·11-s − 0.857·12-s − 5.73·13-s − 3.52·14-s + 1.94·15-s + 16-s + 6.08·17-s − 2.26·18-s + 4.64·19-s − 2.26·20-s + 3.02·21-s − 1.16·22-s − 4.86·23-s − 0.857·24-s + 0.151·25-s − 5.73·26-s + 4.51·27-s − 3.52·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.495·3-s + 0.5·4-s − 1.01·5-s − 0.350·6-s − 1.33·7-s + 0.353·8-s − 0.754·9-s − 0.717·10-s − 0.351·11-s − 0.247·12-s − 1.59·13-s − 0.941·14-s + 0.502·15-s + 0.250·16-s + 1.47·17-s − 0.533·18-s + 1.06·19-s − 0.507·20-s + 0.659·21-s − 0.248·22-s − 1.01·23-s − 0.175·24-s + 0.0302·25-s − 1.12·26-s + 0.869·27-s − 0.665·28-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
\(L(1)\) |
\(\approx\) |
\(0.7539299907\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7539299907\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.857T + 3T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 6.08T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 0.986T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 - 2.45T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 4.65T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.57T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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
−8.106984527166072652142562242724, −7.55119322051448668760587195901, −7.04613437160279802774877970726, −5.99051380676174076673301272877, −5.51739958893504655294286323087, −4.80262496287692892018091348005, −3.56368286049202513112333738562, −3.35042453253719927521707732022, −2.28822002018287489184924135860, −0.42547603994964618186768903357,
0.42547603994964618186768903357, 2.28822002018287489184924135860, 3.35042453253719927521707732022, 3.56368286049202513112333738562, 4.80262496287692892018091348005, 5.51739958893504655294286323087, 5.99051380676174076673301272877, 7.04613437160279802774877970726, 7.55119322051448668760587195901, 8.106984527166072652142562242724