| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s − 18-s − 4·19-s + 21-s − 24-s + 2·26-s + 27-s + 28-s + 6·29-s + 4·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 2·39-s − 6·41-s − 42-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 + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s − 0.937·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.81246910342888, −12.33636554742384, −12.01533055178003, −11.55277423737638, −10.88450174223793, −10.56056310922334, −10.12066613282167, −9.715982485348107, −9.111440824298888, −8.782812049249457, −8.296140685821774, −7.908085136651757, −7.480389118127940, −6.953244992297781, −6.329420082730196, −6.170332924337920, −5.230771829946871, −4.750552607338308, −4.392403257379195, −3.605077939085955, −3.123957150500381, −2.482546887782698, −2.094084121745072, −1.453125232623760, −0.7981162138534166, 0,
0.7981162138534166, 1.453125232623760, 2.094084121745072, 2.482546887782698, 3.123957150500381, 3.605077939085955, 4.392403257379195, 4.750552607338308, 5.230771829946871, 6.170332924337920, 6.329420082730196, 6.953244992297781, 7.480389118127940, 7.908085136651757, 8.296140685821774, 8.782812049249457, 9.111440824298888, 9.715982485348107, 10.12066613282167, 10.56056310922334, 10.88450174223793, 11.55277423737638, 12.01533055178003, 12.33636554742384, 12.81246910342888
