| L(s) = 1 | + 4·3-s − 5·5-s + 7·7-s − 11·9-s − 20·11-s + 10·13-s − 20·15-s − 14·17-s − 12·19-s + 28·21-s + 104·23-s + 25·25-s − 152·27-s + 122·29-s + 224·31-s − 80·33-s − 35·35-s − 158·37-s + 40·39-s + 378·41-s − 404·43-s + 55·45-s + 112·47-s + 49·49-s − 56·51-s − 270·53-s + 100·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.447·5-s + 0.377·7-s − 0.407·9-s − 0.548·11-s + 0.213·13-s − 0.344·15-s − 0.199·17-s − 0.144·19-s + 0.290·21-s + 0.942·23-s + 1/5·25-s − 1.08·27-s + 0.781·29-s + 1.29·31-s − 0.422·33-s − 0.169·35-s − 0.702·37-s + 0.164·39-s + 1.43·41-s − 1.43·43-s + 0.182·45-s + 0.347·47-s + 1/7·49-s − 0.153·51-s − 0.699·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 12 T + p^{3} T^{2} \) |
| 23 | \( 1 - 104 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 404 T + p^{3} T^{2} \) |
| 47 | \( 1 - 112 T + p^{3} T^{2} \) |
| 53 | \( 1 + 270 T + p^{3} T^{2} \) |
| 59 | \( 1 + 324 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 + 156 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 102 T + p^{3} T^{2} \) |
| 79 | \( 1 + 912 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1068 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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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
−8.401396749705539678280364072113, −7.71072616451701854255082284213, −6.90420882731457873032848783538, −5.93265427519409831309417099220, −4.98390944568426141396845394694, −4.20903758782323131532550554053, −3.12473531118455861327321136478, −2.57931620538888695612279615245, −1.31029344486531402605357430268, 0,
1.31029344486531402605357430268, 2.57931620538888695612279615245, 3.12473531118455861327321136478, 4.20903758782323131532550554053, 4.98390944568426141396845394694, 5.93265427519409831309417099220, 6.90420882731457873032848783538, 7.71072616451701854255082284213, 8.401396749705539678280364072113
