| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 4·13-s − 4·14-s − 2·15-s + 16-s − 17-s + 18-s − 8·19-s − 2·20-s − 4·21-s − 22-s + 24-s − 25-s − 4·26-s + 27-s − 4·28-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.447·20-s − 0.872·21-s − 0.213·22-s + 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.755·28-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−19.59946535758159, −19.19184181452814, −18.63855104984210, −17.23211452279612, −16.70976435431377, −15.86848459236044, −15.22132697512076, −15.01955523185509, −13.82292427833942, −13.28778089360189, −12.44958587510269, −12.18536151572100, −11.05285942050468, −10.15695970899537, −9.560427624413729, −8.435750833845311, −7.743686781990299, −6.747549039737825, −6.265116754929520, −4.802770853514411, −4.068044499824602, −3.142984048202357, −2.358799966264125, 0,
2.358799966264125, 3.142984048202357, 4.068044499824602, 4.802770853514411, 6.265116754929520, 6.747549039737825, 7.743686781990299, 8.435750833845311, 9.560427624413729, 10.15695970899537, 11.05285942050468, 12.18536151572100, 12.44958587510269, 13.28778089360189, 13.82292427833942, 15.01955523185509, 15.22132697512076, 15.86848459236044, 16.70976435431377, 17.23211452279612, 18.63855104984210, 19.19184181452814, 19.59946535758159
