L(s) = 1 | − 2-s + 3·3-s − 4-s − 3·6-s + 7-s + 3·8-s + 6·9-s − 3·12-s − 2·13-s − 14-s − 16-s + 3·17-s − 6·18-s + 4·19-s + 3·21-s + 6·23-s + 9·24-s + 2·26-s + 9·27-s − 28-s + 6·29-s + 9·31-s − 5·32-s − 3·34-s − 6·36-s − 11·37-s − 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.22·6-s + 0.377·7-s + 1.06·8-s + 2·9-s − 0.866·12-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 1.41·18-s + 0.917·19-s + 0.654·21-s + 1.25·23-s + 1.83·24-s + 0.392·26-s + 1.73·27-s − 0.188·28-s + 1.11·29-s + 1.61·31-s − 0.883·32-s − 0.514·34-s − 36-s − 1.80·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.465643442\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.465643442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.59945067886414, −14.96139583700640, −14.30867884396287, −14.00463180151906, −13.73058886242291, −13.00976425202729, −12.39135631823697, −11.92778739194414, −10.81757205707008, −10.45019074221659, −9.690124378170856, −9.476121293840296, −8.782428650958999, −8.480952231141949, −7.737379291559303, −7.508294294509193, −6.907505900082323, −5.798947523094228, −4.803125610780088, −4.586116933363559, −3.647656473130600, −3.011201156760775, −2.431103481695243, −1.413432308308899, −0.8802467038996150,
0.8802467038996150, 1.413432308308899, 2.431103481695243, 3.011201156760775, 3.647656473130600, 4.586116933363559, 4.803125610780088, 5.798947523094228, 6.907505900082323, 7.508294294509193, 7.737379291559303, 8.480952231141949, 8.782428650958999, 9.476121293840296, 9.690124378170856, 10.45019074221659, 10.81757205707008, 11.92778739194414, 12.39135631823697, 13.00976425202729, 13.73058886242291, 14.00463180151906, 14.30867884396287, 14.96139583700640, 15.59945067886414