L(s) = 1 | + 2.50·2-s + 0.339·3-s + 4.27·4-s − 5-s + 0.850·6-s − 1.59·7-s + 5.70·8-s − 2.88·9-s − 2.50·10-s − 11-s + 1.45·12-s − 4.84·13-s − 3.99·14-s − 0.339·15-s + 5.73·16-s + 2.08·17-s − 7.22·18-s − 0.364·19-s − 4.27·20-s − 0.540·21-s − 2.50·22-s − 4.64·23-s + 1.93·24-s + 25-s − 12.1·26-s − 1.99·27-s − 6.81·28-s + ⋯ |
L(s) = 1 | + 1.77·2-s + 0.195·3-s + 2.13·4-s − 0.447·5-s + 0.347·6-s − 0.602·7-s + 2.01·8-s − 0.961·9-s − 0.792·10-s − 0.301·11-s + 0.419·12-s − 1.34·13-s − 1.06·14-s − 0.0876·15-s + 1.43·16-s + 0.505·17-s − 1.70·18-s − 0.0835·19-s − 0.956·20-s − 0.117·21-s − 0.534·22-s − 0.968·23-s + 0.395·24-s + 0.200·25-s − 2.37·26-s − 0.384·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 0.339T + 3T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 + 0.364T + 19T^{2} \) |
| 23 | \( 1 + 4.64T + 23T^{2} \) |
| 29 | \( 1 + 0.197T + 29T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + 9.62T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 5.92T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.67402453146206573164395944972, −7.26161881625975349714019610856, −6.36806850615867450286088474520, −5.58833193986113703395301643104, −5.17605557146263104659551134813, −4.16427338008234700383399488674, −3.52296995358616821500939605125, −2.78474374086109348696610713255, −2.10295775404129619997091546720, 0,
2.10295775404129619997091546720, 2.78474374086109348696610713255, 3.52296995358616821500939605125, 4.16427338008234700383399488674, 5.17605557146263104659551134813, 5.58833193986113703395301643104, 6.36806850615867450286088474520, 7.26161881625975349714019610856, 7.67402453146206573164395944972