L(s) = 1 | − 2-s + 1.72·3-s + 4-s − 5-s − 1.72·6-s − 1.94·7-s − 8-s − 0.0274·9-s + 10-s + 1.83·11-s + 1.72·12-s − 2.05·13-s + 1.94·14-s − 1.72·15-s + 16-s + 1.66·17-s + 0.0274·18-s + 1.54·19-s − 20-s − 3.35·21-s − 1.83·22-s + 0.544·23-s − 1.72·24-s + 25-s + 2.05·26-s − 5.21·27-s − 1.94·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.995·3-s + 0.5·4-s − 0.447·5-s − 0.703·6-s − 0.735·7-s − 0.353·8-s − 0.00913·9-s + 0.316·10-s + 0.554·11-s + 0.497·12-s − 0.568·13-s + 0.519·14-s − 0.445·15-s + 0.250·16-s + 0.402·17-s + 0.00646·18-s + 0.353·19-s − 0.223·20-s − 0.731·21-s − 0.391·22-s + 0.113·23-s − 0.351·24-s + 0.200·25-s + 0.402·26-s − 1.00·27-s − 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 0.544T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 + 8.02T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 5.10T + 83T^{2} \) |
| 89 | \( 1 - 0.564T + 89T^{2} \) |
| 97 | \( 1 + 5.43T + 97T^{2} \) |
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
−8.123404022993115673781333244535, −7.61089342630755579544395547276, −6.79912094810344958750562081255, −6.13978902767488115479052318310, −5.05949688379135110524027083715, −3.95286089836457286264965169109, −3.17920460371468431714458034445, −2.61842058068853548103788715576, −1.41062629219028943355932281330, 0,
1.41062629219028943355932281330, 2.61842058068853548103788715576, 3.17920460371468431714458034445, 3.95286089836457286264965169109, 5.05949688379135110524027083715, 6.13978902767488115479052318310, 6.79912094810344958750562081255, 7.61089342630755579544395547276, 8.123404022993115673781333244535