| L(s) = 1 | − 2-s − 4-s − 4·5-s − 2·7-s + 3·8-s + 4·10-s − 11-s − 2·13-s + 2·14-s − 16-s + 2·17-s − 6·19-s + 4·20-s + 22-s + 4·23-s + 11·25-s + 2·26-s + 2·28-s − 6·29-s + 4·31-s − 5·32-s − 2·34-s + 8·35-s − 6·37-s + 6·38-s − 12·40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.755·7-s + 1.06·8-s + 1.26·10-s − 0.301·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s + 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s + 1.35·35-s − 0.986·37-s + 0.973·38-s − 1.89·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.13385760545445737079129373817, −12.34290146894479717918136562556, −11.12010949851048793378147736774, −10.08905583189961630959556424585, −8.821913560068405274367451708628, −7.961683082550188402583017938038, −6.97386872789196147048059276608, −4.78797800594017334892308373003, −3.54703633209054620311690270016, 0,
3.54703633209054620311690270016, 4.78797800594017334892308373003, 6.97386872789196147048059276608, 7.961683082550188402583017938038, 8.821913560068405274367451708628, 10.08905583189961630959556424585, 11.12010949851048793378147736774, 12.34290146894479717918136562556, 13.13385760545445737079129373817
