| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s − 18-s + 8·19-s − 20-s + 4·21-s − 4·22-s + 24-s + 25-s − 2·26-s − 27-s − 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−19.94446274253394, −19.73314027311207, −18.75479320812171, −18.41460904583648, −17.51409053995308, −16.75768072247060, −16.20475116312069, −15.84623980443286, −14.96640541928155, −13.97749055769794, −13.05451746557234, −12.43159026281796, −11.52887185980547, −11.18222396378824, −10.04105229915318, −9.435621590245950, −8.909761485587000, −7.639715523538763, −6.891214434606250, −6.307205494307446, −5.368070285372935, −3.848181817948898, −3.225410955332902, −1.429541736276097, 0,
1.429541736276097, 3.225410955332902, 3.848181817948898, 5.368070285372935, 6.307205494307446, 6.891214434606250, 7.639715523538763, 8.909761485587000, 9.435621590245950, 10.04105229915318, 11.18222396378824, 11.52887185980547, 12.43159026281796, 13.05451746557234, 13.97749055769794, 14.96640541928155, 15.84623980443286, 16.20475116312069, 16.75768072247060, 17.51409053995308, 18.41460904583648, 18.75479320812171, 19.73314027311207, 19.94446274253394
