| L(s) = 1 | + 2-s − 4-s + 5-s − 4·7-s − 3·8-s + 10-s − 5·11-s + 13-s − 4·14-s − 16-s + 5·19-s − 20-s − 5·22-s + 3·23-s + 25-s + 26-s + 4·28-s + 3·29-s − 3·31-s + 5·32-s − 4·35-s + 5·37-s + 5·38-s − 3·40-s − 2·41-s + 3·43-s + 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s + 0.316·10-s − 1.50·11-s + 0.277·13-s − 1.06·14-s − 1/4·16-s + 1.14·19-s − 0.223·20-s − 1.06·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.557·29-s − 0.538·31-s + 0.883·32-s − 0.676·35-s + 0.821·37-s + 0.811·38-s − 0.474·40-s − 0.312·41-s + 0.457·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.41389316655407, −12.93246755753813, −12.73208253161868, −12.44958888678550, −11.47180054575302, −11.33744154700499, −10.43284805023672, −9.963833299818722, −9.860157411376042, −9.122886526129186, −8.891861231328403, −8.136553096373351, −7.660069592157712, −7.029262781460099, −6.497088845668967, −5.956407037258877, −5.614052111234511, −5.080046717233609, −4.634429930447681, −3.939746768394228, −3.151428376896881, −3.068040841823159, −2.573068434361524, −1.540749751115324, −0.6455307599774835, 0,
0.6455307599774835, 1.540749751115324, 2.573068434361524, 3.068040841823159, 3.151428376896881, 3.939746768394228, 4.634429930447681, 5.080046717233609, 5.614052111234511, 5.956407037258877, 6.497088845668967, 7.029262781460099, 7.660069592157712, 8.136553096373351, 8.891861231328403, 9.122886526129186, 9.860157411376042, 9.963833299818722, 10.43284805023672, 11.33744154700499, 11.47180054575302, 12.44958888678550, 12.73208253161868, 12.93246755753813, 13.41389316655407
