| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s + 2·11-s − 12-s + 2·13-s − 15-s − 16-s − 4·17-s + 18-s + 20-s + 2·22-s − 23-s − 3·24-s + 25-s + 2·26-s + 27-s + 29-s − 30-s − 2·31-s + 5·32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.223·20-s + 0.426·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.185·29-s − 0.182·30-s − 0.359·31-s + 0.883·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.97253854161408, −16.01179480526197, −15.74757207708826, −15.05369772001261, −14.47063841160399, −14.19631790009649, −13.32642379498414, −13.17363669767037, −12.38409738061395, −11.89770852496971, −11.15250393104525, −10.63733981883944, −9.492142936172275, −9.367147370219083, −8.551250666566472, −8.126507112848970, −7.322461683746430, −6.419173156585503, −6.080104760162119, −4.951237660074223, −4.510893000303053, −3.776916385125830, −3.310383577653069, −2.393326952489312, −1.287333937066845, 0,
1.287333937066845, 2.393326952489312, 3.310383577653069, 3.776916385125830, 4.510893000303053, 4.951237660074223, 6.080104760162119, 6.419173156585503, 7.322461683746430, 8.126507112848970, 8.551250666566472, 9.367147370219083, 9.492142936172275, 10.63733981883944, 11.15250393104525, 11.89770852496971, 12.38409738061395, 13.17363669767037, 13.32642379498414, 14.19631790009649, 14.47063841160399, 15.05369772001261, 15.74757207708826, 16.01179480526197, 16.97253854161408
