L(s) = 1 | − 2-s − 4-s + 2·5-s + 4·7-s + 3·8-s − 2·10-s − 11-s − 2·13-s − 4·14-s − 16-s + 2·17-s − 2·20-s + 22-s − 8·23-s − 25-s + 2·26-s − 4·28-s + 6·29-s − 8·31-s − 5·32-s − 2·34-s + 8·35-s + 6·37-s + 6·40-s + 2·41-s + 44-s + 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.06·8-s − 0.632·10-s − 0.301·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 1.35·35-s + 0.986·37-s + 0.948·40-s + 0.312·41-s + 0.150·44-s + 1.17·46-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
\(L(1)\) |
\(\approx\) |
\(0.7923074038\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7923074038\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.12737143370045703342633485701, −12.97885466615636407297758144806, −11.61114553747330884538582571440, −10.39134307397473772625111872930, −9.637990407765945223554197990480, −8.389268452519734683124359977272, −7.60731726518247812897599595849, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −1.82001999803537272093437549398,
1.82001999803537272093437549398, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 7.60731726518247812897599595849, 8.389268452519734683124359977272, 9.637990407765945223554197990480, 10.39134307397473772625111872930, 11.61114553747330884538582571440, 12.97885466615636407297758144806, 14.12737143370045703342633485701