| L(s) = 1 | + 5-s + 3·7-s + 5·11-s − 4·13-s + 3·17-s + 19-s + 8·23-s − 4·25-s + 2·29-s − 4·31-s + 3·35-s + 10·37-s − 10·41-s − 43-s − 47-s + 2·49-s + 4·53-s + 5·55-s + 6·59-s − 13·61-s − 4·65-s + 12·67-s + 2·71-s + 9·73-s + 15·77-s − 8·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.50·11-s − 1.10·13-s + 0.727·17-s + 0.229·19-s + 1.66·23-s − 4/5·25-s + 0.371·29-s − 0.718·31-s + 0.507·35-s + 1.64·37-s − 1.56·41-s − 0.152·43-s − 0.145·47-s + 2/7·49-s + 0.549·53-s + 0.674·55-s + 0.781·59-s − 1.66·61-s − 0.496·65-s + 1.46·67-s + 0.237·71-s + 1.05·73-s + 1.70·77-s − 0.900·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564404558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564404558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.880140284265606450558573740236, −8.064901679875409643106905234825, −7.28807747269950922759271760406, −6.64017519777900431546557110668, −5.61259928782815339019051631078, −4.96316680572620019147167415239, −4.17517686433721218560152257029, −3.09871623117322073739701851963, −1.94226086883320328992152894180, −1.10108411325297450707135792130,
1.10108411325297450707135792130, 1.94226086883320328992152894180, 3.09871623117322073739701851963, 4.17517686433721218560152257029, 4.96316680572620019147167415239, 5.61259928782815339019051631078, 6.64017519777900431546557110668, 7.28807747269950922759271760406, 8.064901679875409643106905234825, 8.880140284265606450558573740236
