| L(s) = 1 | + 0.871·3-s − 2.41·7-s − 2.23·9-s − 2.51·11-s + 3.34·13-s + 2.99·17-s − 6.82·19-s − 2.10·21-s + 7.12·23-s − 4.56·27-s + 5.63·29-s + 7.29·31-s − 2.19·33-s − 0.162·37-s + 2.91·39-s + 41-s − 10.1·43-s − 8.46·47-s − 1.16·49-s + 2.61·51-s − 0.116·53-s − 5.94·57-s + 9.40·59-s + 15.0·61-s + 5.41·63-s − 2.21·67-s + 6.20·69-s + ⋯ |
| L(s) = 1 | + 0.503·3-s − 0.913·7-s − 0.746·9-s − 0.757·11-s + 0.927·13-s + 0.726·17-s − 1.56·19-s − 0.459·21-s + 1.48·23-s − 0.879·27-s + 1.04·29-s + 1.31·31-s − 0.381·33-s − 0.0267·37-s + 0.466·39-s + 0.156·41-s − 1.55·43-s − 1.23·47-s − 0.165·49-s + 0.365·51-s − 0.0159·53-s − 0.787·57-s + 1.22·59-s + 1.93·61-s + 0.681·63-s − 0.270·67-s + 0.747·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.687667813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687667813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.871T + 3T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 + 0.162T + 37T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 + 0.116T + 53T^{2} \) |
| 59 | \( 1 - 9.40T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 - 0.0998T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
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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
−8.370498131271876560727382353476, −8.020547909463293993396101923733, −6.72114356408660426199205679058, −6.43072335109238145628024481038, −5.47818264958750916383347822863, −4.68766572972614726176087734398, −3.51279158036349162735429655766, −3.06598251614843795617932651555, −2.19038128470589948991557568276, −0.70010579670781711297308703865,
0.70010579670781711297308703865, 2.19038128470589948991557568276, 3.06598251614843795617932651555, 3.51279158036349162735429655766, 4.68766572972614726176087734398, 5.47818264958750916383347822863, 6.43072335109238145628024481038, 6.72114356408660426199205679058, 8.020547909463293993396101923733, 8.370498131271876560727382353476
