| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s + 4·19-s + 2·20-s + 4·22-s − 25-s − 2·26-s − 28-s − 2·29-s + 5·32-s − 2·35-s − 6·37-s + 4·38-s + 6·40-s + 2·41-s − 4·43-s − 4·44-s + 49-s − 50-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 + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.883·32-s − 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.95066628362837, −15.23694510896951, −14.99347282634105, −14.36447229976918, −13.87339294285049, −13.53695528219071, −12.56693599249134, −12.24393215700795, −11.79650828416118, −11.32659985566129, −10.62940818322475, −9.673888266405059, −9.367959610241320, −8.745853962455529, −8.013096519700251, −7.587425002085384, −6.775282039674580, −6.209136998053063, −5.346370306353391, −4.876916872726395, −4.229666405576952, −3.615704252971583, −3.214406595570456, −2.060674137441135, −1.028751621894537, 0,
1.028751621894537, 2.060674137441135, 3.214406595570456, 3.615704252971583, 4.229666405576952, 4.876916872726395, 5.346370306353391, 6.209136998053063, 6.775282039674580, 7.587425002085384, 8.013096519700251, 8.745853962455529, 9.367959610241320, 9.673888266405059, 10.62940818322475, 11.32659985566129, 11.79650828416118, 12.24393215700795, 12.56693599249134, 13.53695528219071, 13.87339294285049, 14.36447229976918, 14.99347282634105, 15.23694510896951, 15.95066628362837
