| L(s) = 1 | + 2-s − 4-s − 5-s − 4·7-s − 3·8-s − 10-s + 2·11-s − 13-s − 4·14-s − 16-s + 4·19-s + 20-s + 2·22-s − 8·23-s + 25-s − 26-s + 4·28-s + 6·29-s − 8·31-s + 5·32-s + 4·35-s + 4·37-s + 4·38-s + 3·40-s − 2·41-s − 6·43-s − 2·44-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 + 0.603·11-s − 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.917·19-s + 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s + 0.676·35-s + 0.657·37-s + 0.648·38-s + 0.474·40-s − 0.312·41-s − 0.914·43-s − 0.301·44-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 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.39554158953615, −13.07010748555077, −12.45440519307054, −12.08786031934892, −11.97604250124891, −11.20930908095726, −10.60981120337013, −9.892229242877289, −9.720246208456059, −9.224202030571422, −8.844340728250512, −7.926468830972183, −7.902126875416973, −6.872572750054415, −6.603614073216132, −6.113052068651486, −5.574849780545319, −5.057023640843262, −4.401096057951106, −3.905219581889033, −3.474878698009838, −3.090939750653262, −2.440876065507553, −1.524369773766708, −0.5861305577858153, 0,
0.5861305577858153, 1.524369773766708, 2.440876065507553, 3.090939750653262, 3.474878698009838, 3.905219581889033, 4.401096057951106, 5.057023640843262, 5.574849780545319, 6.113052068651486, 6.603614073216132, 6.872572750054415, 7.902126875416973, 7.926468830972183, 8.844340728250512, 9.224202030571422, 9.720246208456059, 9.892229242877289, 10.60981120337013, 11.20930908095726, 11.97604250124891, 12.08786031934892, 12.45440519307054, 13.07010748555077, 13.39554158953615
