L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s − 7·13-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s + 25-s − 7·26-s − 9·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s − 1.94·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.37·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6269264199\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6269264199\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.84040877228253, −14.20898241484622, −13.67734230965608, −13.03249272148460, −12.60724874665118, −12.15143843774710, −11.73115347145848, −11.28843573250233, −10.55317334807268, −10.33489868879431, −9.767235370715473, −9.172561336511538, −7.974244952020230, −7.781806894488222, −6.793758275259766, −6.614091592814273, −6.032368592091764, −5.197791545851071, −4.993428439952617, −4.665097664153090, −3.847837084831771, −2.792405009766381, −2.139205702698853, −1.483686724818508, −0.2750246464237492,
0.2750246464237492, 1.483686724818508, 2.139205702698853, 2.792405009766381, 3.847837084831771, 4.665097664153090, 4.993428439952617, 5.197791545851071, 6.032368592091764, 6.614091592814273, 6.793758275259766, 7.781806894488222, 7.974244952020230, 9.172561336511538, 9.767235370715473, 10.33489868879431, 10.55317334807268, 11.28843573250233, 11.73115347145848, 12.15143843774710, 12.60724874665118, 13.03249272148460, 13.67734230965608, 14.20898241484622, 14.84040877228253