L(s) = 1 | − 2-s + 3.34·3-s + 4-s − 5-s − 3.34·6-s + 2.01·7-s − 8-s + 8.20·9-s + 10-s − 11-s + 3.34·12-s + 1.48·13-s − 2.01·14-s − 3.34·15-s + 16-s − 4.62·17-s − 8.20·18-s + 4.61·19-s − 20-s + 6.74·21-s + 22-s − 0.510·23-s − 3.34·24-s + 25-s − 1.48·26-s + 17.4·27-s + 2.01·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.93·3-s + 0.5·4-s − 0.447·5-s − 1.36·6-s + 0.761·7-s − 0.353·8-s + 2.73·9-s + 0.316·10-s − 0.301·11-s + 0.966·12-s + 0.413·13-s − 0.538·14-s − 0.864·15-s + 0.250·16-s − 1.12·17-s − 1.93·18-s + 1.05·19-s − 0.223·20-s + 1.47·21-s + 0.213·22-s − 0.106·23-s − 0.683·24-s + 0.200·25-s − 0.292·26-s + 3.35·27-s + 0.380·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.224874007\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.224874007\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 + 3.21T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 47 | \( 1 - 0.687T + 47T^{2} \) |
| 53 | \( 1 + 9.90T + 53T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.204659695156888659066808035459, −7.970512350619944270647004417933, −7.17461397474143873628275417033, −6.60426892759580654212855818648, −5.16568216698011169267595819171, −4.31174419066932490303317764722, −3.55726486485835944807717513774, −2.73325079761686572739377972141, −2.01649473733527147337702941008, −1.07556726665076332465419857810,
1.07556726665076332465419857810, 2.01649473733527147337702941008, 2.73325079761686572739377972141, 3.55726486485835944807717513774, 4.31174419066932490303317764722, 5.16568216698011169267595819171, 6.60426892759580654212855818648, 7.17461397474143873628275417033, 7.970512350619944270647004417933, 8.204659695156888659066808035459