L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 3·11-s − 12-s + 4·13-s + 14-s + 16-s + 18-s − 21-s − 3·22-s + 3·23-s − 24-s + 4·26-s − 27-s + 28-s − 9·29-s + 7·31-s + 32-s + 3·33-s + 36-s − 2·37-s − 4·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.218·21-s − 0.639·22-s + 0.625·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.67·29-s + 1.25·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 0.328·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.68173746306996, −12.38330007677832, −11.76293324564322, −11.25911484395916, −11.03829907622036, −10.68454589558062, −10.18911163921150, −9.558298043267925, −9.174554252194365, −8.448012336162062, −8.098644620129216, −7.624323967604494, −7.056474178967886, −6.706625858811746, −6.007027331010091, −5.713034210568125, −5.297528515289373, −4.804783727230742, −4.242954984171985, −3.781894055716183, −3.309520061395302, −2.464680650517774, −2.253279993437613, −1.283645244366672, −0.9382040970623845, 0,
0.9382040970623845, 1.283645244366672, 2.253279993437613, 2.464680650517774, 3.309520061395302, 3.781894055716183, 4.242954984171985, 4.804783727230742, 5.297528515289373, 5.713034210568125, 6.007027331010091, 6.706625858811746, 7.056474178967886, 7.624323967604494, 8.098644620129216, 8.448012336162062, 9.174554252194365, 9.558298043267925, 10.18911163921150, 10.68454589558062, 11.03829907622036, 11.25911484395916, 11.76293324564322, 12.38330007677832, 12.68173746306996