| L(s) = 1 | − 2·3-s + 5·5-s − 6·7-s − 23·9-s + 60·11-s − 50·13-s − 10·15-s − 30·17-s + 40·19-s + 12·21-s − 178·23-s + 25·25-s + 100·27-s − 166·29-s − 20·31-s − 120·33-s − 30·35-s − 10·37-s + 100·39-s − 250·41-s + 142·43-s − 115·45-s − 214·47-s − 307·49-s + 60·51-s − 490·53-s + 300·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.447·5-s − 0.323·7-s − 0.851·9-s + 1.64·11-s − 1.06·13-s − 0.172·15-s − 0.428·17-s + 0.482·19-s + 0.124·21-s − 1.61·23-s + 1/5·25-s + 0.712·27-s − 1.06·29-s − 0.115·31-s − 0.633·33-s − 0.144·35-s − 0.0444·37-s + 0.410·39-s − 0.952·41-s + 0.503·43-s − 0.380·45-s − 0.664·47-s − 0.895·49-s + 0.164·51-s − 1.26·53-s + 0.735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 178 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 10 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 - 142 T + p^{3} T^{2} \) |
| 47 | \( 1 + 214 T + p^{3} T^{2} \) |
| 53 | \( 1 + 490 T + p^{3} T^{2} \) |
| 59 | \( 1 + 800 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 774 T + p^{3} T^{2} \) |
| 71 | \( 1 + 100 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 982 T + p^{3} T^{2} \) |
| 89 | \( 1 - 874 T + p^{3} T^{2} \) |
| 97 | \( 1 + 310 T + p^{3} T^{2} \) |
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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
−10.80955299354140111544658257243, −9.632865999917779694518672265143, −9.137331091314926591834385597847, −7.82969450661548265181638527568, −6.58691942210092228967766333516, −5.93359824859044477961355494389, −4.70877250257756646000235120019, −3.35755090485553896551599009385, −1.83584664853183369017185571194, 0,
1.83584664853183369017185571194, 3.35755090485553896551599009385, 4.70877250257756646000235120019, 5.93359824859044477961355494389, 6.58691942210092228967766333516, 7.82969450661548265181638527568, 9.137331091314926591834385597847, 9.632865999917779694518672265143, 10.80955299354140111544658257243
