| L(s) = 1 | − 1.32·2-s − 0.233·4-s + 3.11·7-s + 2.96·8-s + 4.02·11-s + 6.45·13-s − 4.14·14-s − 3.47·16-s − 0.628·17-s − 8.46·19-s − 5.34·22-s + 4.61·23-s − 8.57·26-s − 0.729·28-s + 1.58·29-s − 0.511·31-s − 1.31·32-s + 0.835·34-s − 37-s + 11.2·38-s + 9.85·41-s − 5.71·43-s − 0.941·44-s − 6.12·46-s + 2.47·47-s + 2.71·49-s − 1.50·52-s + ⋯ |
| L(s) = 1 | − 0.939·2-s − 0.116·4-s + 1.17·7-s + 1.04·8-s + 1.21·11-s + 1.78·13-s − 1.10·14-s − 0.869·16-s − 0.152·17-s − 1.94·19-s − 1.13·22-s + 0.961·23-s − 1.68·26-s − 0.137·28-s + 0.293·29-s − 0.0918·31-s − 0.232·32-s + 0.143·34-s − 0.164·37-s + 1.82·38-s + 1.53·41-s − 0.871·43-s − 0.141·44-s − 0.903·46-s + 0.361·47-s + 0.388·49-s − 0.209·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655511785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655511785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 - 6.45T + 13T^{2} \) |
| 17 | \( 1 + 0.628T + 17T^{2} \) |
| 19 | \( 1 + 8.46T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 + 0.511T + 31T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 3.99T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 - 9.87T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−8.216861979604832305621224533859, −7.18188230304859283360130284163, −6.60447323298861445751190281579, −5.84851073987900581343953423277, −4.90501593397285165793173902245, −4.15221668383087956334649951724, −3.79246701598885184233085181716, −2.26634014894010499707920485522, −1.45081650970951734405380678143, −0.841177096520293641013278362841,
0.841177096520293641013278362841, 1.45081650970951734405380678143, 2.26634014894010499707920485522, 3.79246701598885184233085181716, 4.15221668383087956334649951724, 4.90501593397285165793173902245, 5.84851073987900581343953423277, 6.60447323298861445751190281579, 7.18188230304859283360130284163, 8.216861979604832305621224533859
