| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·13-s − 14-s + 16-s − 4·17-s − 2·19-s + 20-s + 23-s + 25-s + 4·26-s − 28-s − 10·29-s − 6·31-s + 32-s − 4·34-s − 35-s − 6·37-s − 2·38-s + 40-s + 2·41-s − 4·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 1.85·29-s − 1.07·31-s + 0.176·32-s − 0.685·34-s − 0.169·35-s − 0.986·37-s − 0.324·38-s + 0.158·40-s + 0.312·41-s − 0.609·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.34587806751263, −15.75511766792817, −15.10732715144069, −14.86857502441386, −13.97091479323275, −13.50198021450414, −13.15709893625940, −12.65352563658277, −11.92808261924927, −11.31317594782588, −10.62253289073309, −10.50352078553387, −9.314038420640941, −9.047699078748515, −8.382437672195954, −7.377539534074540, −7.008025950953121, −6.177433263563186, −5.792039014117760, −5.159506905256594, −4.186444700051532, −3.783625230442213, −2.958147813573366, −2.102576887252670, −1.433676572947168, 0,
1.433676572947168, 2.102576887252670, 2.958147813573366, 3.783625230442213, 4.186444700051532, 5.159506905256594, 5.792039014117760, 6.177433263563186, 7.008025950953121, 7.377539534074540, 8.382437672195954, 9.047699078748515, 9.314038420640941, 10.50352078553387, 10.62253289073309, 11.31317594782588, 11.92808261924927, 12.65352563658277, 13.15709893625940, 13.50198021450414, 13.97091479323275, 14.86857502441386, 15.10732715144069, 15.75511766792817, 16.34587806751263
