| L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s + 8·7-s − 8·8-s + 10·10-s − 18·11-s + 8·13-s − 16·14-s + 16·16-s − 15·17-s + 23·19-s − 20·20-s + 36·22-s − 63·23-s + 25·25-s − 16·26-s + 32·28-s − 156·29-s − 85·31-s − 32·32-s + 30·34-s − 40·35-s + 74·37-s − 46·38-s + 40·40-s − 246·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.431·7-s − 0.353·8-s + 0.316·10-s − 0.493·11-s + 0.170·13-s − 0.305·14-s + 1/4·16-s − 0.214·17-s + 0.277·19-s − 0.223·20-s + 0.348·22-s − 0.571·23-s + 1/5·25-s − 0.120·26-s + 0.215·28-s − 0.998·29-s − 0.492·31-s − 0.176·32-s + 0.151·34-s − 0.193·35-s + 0.328·37-s − 0.196·38-s + 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 15 T + p^{3} T^{2} \) |
| 19 | \( 1 - 23 T + p^{3} T^{2} \) |
| 23 | \( 1 + 63 T + p^{3} T^{2} \) |
| 29 | \( 1 + 156 T + p^{3} T^{2} \) |
| 31 | \( 1 + 85 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 190 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 - 177 T + p^{3} T^{2} \) |
| 59 | \( 1 + 792 T + p^{3} T^{2} \) |
| 61 | \( 1 + 907 T + p^{3} T^{2} \) |
| 67 | \( 1 + 322 T + p^{3} T^{2} \) |
| 71 | \( 1 - 270 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 83 | \( 1 - 771 T + p^{3} T^{2} \) |
| 89 | \( 1 - 198 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1192 T + p^{3} T^{2} \) |
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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
−11.00564029049174546806247344932, −10.06282641045245198233424185111, −9.021314292783839211692889492912, −8.067391729731292506121444314228, −7.35915803419004134481334040810, −6.09539103160029208356992626184, −4.79983914086081194830581841482, −3.32453212744109124666567201438, −1.74934453523367868552826874302, 0,
1.74934453523367868552826874302, 3.32453212744109124666567201438, 4.79983914086081194830581841482, 6.09539103160029208356992626184, 7.35915803419004134481334040810, 8.067391729731292506121444314228, 9.021314292783839211692889492912, 10.06282641045245198233424185111, 11.00564029049174546806247344932
