| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 4·13-s − 14-s + 16-s + 3·17-s + 18-s − 5·19-s − 21-s − 22-s + 3·23-s + 24-s − 5·25-s + 4·26-s + 27-s − 28-s + 6·29-s + 4·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.213·22-s + 0.625·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.89043733846324, −13.74995400100805, −13.13563224089388, −12.78387681492312, −12.25857575198667, −11.81745091121762, −10.97969236436210, −10.85519687899834, −10.15469299793054, −9.658051842238961, −9.159463804525973, −8.406799855470048, −8.101965320314341, −7.645484856224312, −6.834818482095586, −6.329088213666411, −6.087027514037902, −5.300224113478945, −4.637882297473673, −4.213798025904620, −3.505722534105168, −3.080485773024356, −2.556780908393586, −1.694222677614867, −1.167442066095139, 0,
1.167442066095139, 1.694222677614867, 2.556780908393586, 3.080485773024356, 3.505722534105168, 4.213798025904620, 4.637882297473673, 5.300224113478945, 6.087027514037902, 6.329088213666411, 6.834818482095586, 7.645484856224312, 8.101965320314341, 8.406799855470048, 9.159463804525973, 9.658051842238961, 10.15469299793054, 10.85519687899834, 10.97969236436210, 11.81745091121762, 12.25857575198667, 12.78387681492312, 13.13563224089388, 13.74995400100805, 13.89043733846324
