| L(s) = 1 | + 2·3-s + 7·7-s − 23·9-s − 20·11-s + 20·13-s − 50·17-s − 10·19-s + 14·21-s − 72·23-s − 125·25-s − 100·27-s + 134·29-s − 180·31-s − 40·33-s + 270·37-s + 40·39-s − 250·41-s − 92·43-s − 236·47-s + 49·49-s − 100·51-s − 150·53-s − 20·57-s − 570·59-s + 200·61-s − 161·63-s − 176·67-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.377·7-s − 0.851·9-s − 0.548·11-s + 0.426·13-s − 0.713·17-s − 0.120·19-s + 0.145·21-s − 0.652·23-s − 25-s − 0.712·27-s + 0.858·29-s − 1.04·31-s − 0.211·33-s + 1.19·37-s + 0.164·39-s − 0.952·41-s − 0.326·43-s − 0.732·47-s + 1/7·49-s − 0.274·51-s − 0.388·53-s − 0.0464·57-s − 1.25·59-s + 0.419·61-s − 0.321·63-s − 0.320·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 10 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 236 T + p^{3} T^{2} \) |
| 53 | \( 1 + 150 T + p^{3} T^{2} \) |
| 59 | \( 1 + 570 T + p^{3} T^{2} \) |
| 61 | \( 1 - 200 T + p^{3} T^{2} \) |
| 67 | \( 1 + 176 T + p^{3} T^{2} \) |
| 71 | \( 1 + 640 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 + 882 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1074 T + p^{3} T^{2} \) |
| 97 | \( 1 - 270 T + p^{3} T^{2} \) |
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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
−10.25748498324957504509131370074, −9.208893105052666805852626045420, −8.360064254558890849495236277909, −7.72086062754487402184596787687, −6.39951055015455975759044251402, −5.48064039008474836675410033895, −4.29035064708050804730050201363, −3.05407244377906883084601405659, −1.89295776630798311401572394705, 0,
1.89295776630798311401572394705, 3.05407244377906883084601405659, 4.29035064708050804730050201363, 5.48064039008474836675410033895, 6.39951055015455975759044251402, 7.72086062754487402184596787687, 8.360064254558890849495236277909, 9.208893105052666805852626045420, 10.25748498324957504509131370074
