| L(s) = 1 | − 1.63·2-s + 1.98·3-s + 0.682·4-s + 5-s − 3.25·6-s + 2.15·8-s + 0.944·9-s − 1.63·10-s − 5.85·11-s + 1.35·12-s + 2.92·13-s + 1.98·15-s − 4.89·16-s + 0.243·17-s − 1.54·18-s + 3.39·19-s + 0.682·20-s + 9.58·22-s + 23-s + 4.28·24-s + 25-s − 4.78·26-s − 4.08·27-s + 2.45·29-s − 3.25·30-s − 4.71·31-s + 3.70·32-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 1.14·3-s + 0.341·4-s + 0.447·5-s − 1.32·6-s + 0.763·8-s + 0.314·9-s − 0.517·10-s − 1.76·11-s + 0.391·12-s + 0.810·13-s + 0.512·15-s − 1.22·16-s + 0.0590·17-s − 0.364·18-s + 0.778·19-s + 0.152·20-s + 2.04·22-s + 0.208·23-s + 0.875·24-s + 0.200·25-s − 0.938·26-s − 0.785·27-s + 0.456·29-s − 0.593·30-s − 0.847·31-s + 0.655·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 - 1.98T + 3T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 0.243T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 5.97T + 41T^{2} \) |
| 43 | \( 1 - 0.890T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 + 0.531T + 53T^{2} \) |
| 59 | \( 1 - 6.77T + 59T^{2} \) |
| 61 | \( 1 - 7.39T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 - 1.54T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.138550929007898952021233823614, −7.42818458422092146511288954693, −6.72341793110455462484461142467, −5.49554487922370827089714235049, −5.04233001997744812620304510426, −3.79739190559743820072178421201, −3.00808687418481054258098367858, −2.22910922817147712447291587633, −1.37998755894025123513876666949, 0,
1.37998755894025123513876666949, 2.22910922817147712447291587633, 3.00808687418481054258098367858, 3.79739190559743820072178421201, 5.04233001997744812620304510426, 5.49554487922370827089714235049, 6.72341793110455462484461142467, 7.42818458422092146511288954693, 8.138550929007898952021233823614
