| L(s) = 1 | − 3.73·5-s + 7-s + 5.19·11-s − 3.46·13-s − 4·17-s + 7.92·19-s − 3.73·23-s + 8.92·25-s − 8.92·29-s − 8.46·31-s − 3.73·35-s + 6.46·37-s + 1.19·41-s − 1.46·43-s − 5.46·47-s + 49-s − 8·53-s − 19.3·55-s − 2.53·59-s − 2.92·61-s + 12.9·65-s − 15.4·67-s + 2.66·71-s − 12.3·73-s + 5.19·77-s + 8.39·79-s − 0.928·83-s + ⋯ |
| L(s) = 1 | − 1.66·5-s + 0.377·7-s + 1.56·11-s − 0.960·13-s − 0.970·17-s + 1.81·19-s − 0.778·23-s + 1.78·25-s − 1.65·29-s − 1.52·31-s − 0.630·35-s + 1.06·37-s + 0.186·41-s − 0.223·43-s − 0.797·47-s + 0.142·49-s − 1.09·53-s − 2.61·55-s − 0.330·59-s − 0.374·61-s + 1.60·65-s − 1.88·67-s + 0.315·71-s − 1.45·73-s + 0.592·77-s + 0.944·79-s − 0.101·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 0.928T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.180000771077112947926704660426, −8.116045413433333961712382849044, −7.44099345821775860490515778698, −6.98596344719317743109012914885, −5.74065504275775049181495892484, −4.61136514575065980489563030078, −4.00289156281530574603209504909, −3.17567269939134536186204241946, −1.58307190478424044227411320567, 0,
1.58307190478424044227411320567, 3.17567269939134536186204241946, 4.00289156281530574603209504909, 4.61136514575065980489563030078, 5.74065504275775049181495892484, 6.98596344719317743109012914885, 7.44099345821775860490515778698, 8.116045413433333961712382849044, 9.180000771077112947926704660426
