| L(s) = 1 | − 5-s + 0.527·7-s − 3.11·11-s + 4.11·13-s + 4.39·17-s − 3.70·19-s + 23-s + 25-s + 9.10·29-s + 4.83·31-s − 0.527·35-s + 9.74·37-s − 6.93·41-s − 4.45·43-s − 0.642·47-s − 6.72·49-s + 3.89·53-s + 3.11·55-s − 8.79·59-s − 3.45·61-s − 4.11·65-s − 8.60·67-s − 12.3·71-s − 5.81·73-s − 1.64·77-s − 3.17·79-s − 4.71·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.199·7-s − 0.939·11-s + 1.14·13-s + 1.06·17-s − 0.849·19-s + 0.208·23-s + 0.200·25-s + 1.69·29-s + 0.867·31-s − 0.0891·35-s + 1.60·37-s − 1.08·41-s − 0.680·43-s − 0.0936·47-s − 0.960·49-s + 0.534·53-s + 0.420·55-s − 1.14·59-s − 0.442·61-s − 0.510·65-s − 1.05·67-s − 1.46·71-s − 0.680·73-s − 0.187·77-s − 0.357·79-s − 0.517·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.867060346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867060346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 0.527T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 - 9.74T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 0.642T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 8.60T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−7.910869312935522876290184357871, −7.23454839419772069286180434097, −6.26351507911935154832854680165, −5.92305857251423681245221751532, −4.79253641677560298344890938299, −4.47184033676422710750795483654, −3.32264986205820500148066049159, −2.88445882692766116481773595049, −1.66800571469461125418401929365, −0.69214017199874804944040582976,
0.69214017199874804944040582976, 1.66800571469461125418401929365, 2.88445882692766116481773595049, 3.32264986205820500148066049159, 4.47184033676422710750795483654, 4.79253641677560298344890938299, 5.92305857251423681245221751532, 6.26351507911935154832854680165, 7.23454839419772069286180434097, 7.910869312935522876290184357871
