| L(s) = 1 | + 2-s − 4-s + 2·5-s − 7-s − 3·8-s + 2·10-s + 11-s + 6·13-s − 14-s − 16-s + 2·17-s − 8·19-s − 2·20-s + 22-s − 25-s + 6·26-s + 28-s − 29-s − 4·31-s + 5·32-s + 2·34-s − 2·35-s − 2·37-s − 8·38-s − 6·40-s + 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.185·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s − 0.338·35-s − 0.328·37-s − 1.29·38-s − 0.948·40-s + 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−15.66546266657209, −15.40009467695295, −14.60493403319206, −14.25578919085300, −13.66683285426283, −13.20384980533089, −12.90421886655931, −12.32731968984033, −11.61505899742070, −10.95686205293202, −10.35725202979768, −9.848875699034114, −9.112979468893237, −8.728707142177840, −8.294639684254634, −7.261591009848542, −6.454549072742500, −5.996346423972287, −5.727958245035969, −4.887774742247819, −4.008977676684882, −3.778470359463706, −2.904901129871152, −2.032146372754588, −1.204224267842237, 0,
1.204224267842237, 2.032146372754588, 2.904901129871152, 3.778470359463706, 4.008977676684882, 4.887774742247819, 5.727958245035969, 5.996346423972287, 6.454549072742500, 7.261591009848542, 8.294639684254634, 8.728707142177840, 9.112979468893237, 9.848875699034114, 10.35725202979768, 10.95686205293202, 11.61505899742070, 12.32731968984033, 12.90421886655931, 13.20384980533089, 13.66683285426283, 14.25578919085300, 14.60493403319206, 15.40009467695295, 15.66546266657209
