| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s − 12-s + 2·13-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s + 2·20-s − 21-s − 23-s + 24-s − 25-s − 2·26-s − 27-s + 28-s + 2·29-s + 2·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.75352097620809, −13.40345614744093, −12.84989033998667, −12.20769061932016, −11.81990988955250, −11.40906696846346, −10.81710905285803, −10.27980663190381, −10.06145649608389, −9.564794560471792, −8.897037264559385, −8.480845276538877, −7.990320906897059, −7.389328682796278, −6.752168463367573, −6.392036001168994, −5.857455750380677, −5.292555353489865, −4.939636945273106, −4.067982438774091, −3.468182780286879, −2.750454936735977, −1.977285417364048, −1.547216354053461, −0.9164154620867774, 0,
0.9164154620867774, 1.547216354053461, 1.977285417364048, 2.750454936735977, 3.468182780286879, 4.067982438774091, 4.939636945273106, 5.292555353489865, 5.857455750380677, 6.392036001168994, 6.752168463367573, 7.389328682796278, 7.990320906897059, 8.480845276538877, 8.897037264559385, 9.564794560471792, 10.06145649608389, 10.27980663190381, 10.81710905285803, 11.40906696846346, 11.81990988955250, 12.20769061932016, 12.84989033998667, 13.40345614744093, 13.75352097620809
