| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−8.601405563217285686555412492056, −7.75128918025107996069174544630, −6.96846308061153964022842135026, −6.14836806197630852271087800199, −5.44948790055200400199659619793, −4.40438572519139842235840976944, −3.88341084047931408724670128105, −2.27571844436291562708477831570, −1.86485531322756247966908211949, 0,
1.86485531322756247966908211949, 2.27571844436291562708477831570, 3.88341084047931408724670128105, 4.40438572519139842235840976944, 5.44948790055200400199659619793, 6.14836806197630852271087800199, 6.96846308061153964022842135026, 7.75128918025107996069174544630, 8.601405563217285686555412492056
