| L(s) = 1 | − 5-s + 1.73·7-s − 1.73·11-s + 3·13-s − 4·17-s + 6.92·19-s − 8.66·23-s − 4·25-s + 29-s − 5.19·31-s − 1.73·35-s + 8·37-s − 5·41-s − 8.66·43-s + 12.1·47-s − 4·49-s − 8·53-s + 1.73·55-s − 1.73·59-s + 7·61-s − 3·65-s − 8.66·67-s + 3.46·71-s − 12·73-s − 2.99·77-s + 5.19·79-s + 8.66·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.654·7-s − 0.522·11-s + 0.832·13-s − 0.970·17-s + 1.58·19-s − 1.80·23-s − 0.800·25-s + 0.185·29-s − 0.933·31-s − 0.292·35-s + 1.31·37-s − 0.780·41-s − 1.32·43-s + 1.76·47-s − 0.571·49-s − 1.09·53-s + 0.233·55-s − 0.225·59-s + 0.896·61-s − 0.372·65-s − 1.05·67-s + 0.411·71-s − 1.40·73-s − 0.341·77-s + 0.584·79-s + 0.950·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−7.87952044240999256766427664650, −7.35405543094406606722650232925, −6.33797914767129957760086287711, −5.68785511731878536963599555958, −4.89640991614013405661000721394, −4.09011670481825052910150333744, −3.42151429483412598455337828239, −2.30903502624702498703141712776, −1.40070458817986799643669663738, 0,
1.40070458817986799643669663738, 2.30903502624702498703141712776, 3.42151429483412598455337828239, 4.09011670481825052910150333744, 4.89640991614013405661000721394, 5.68785511731878536963599555958, 6.33797914767129957760086287711, 7.35405543094406606722650232925, 7.87952044240999256766427664650
