| L(s) = 1 | − 0.879·2-s − 1.22·4-s − 0.879·5-s − 4.41·7-s + 2.83·8-s + 0.773·10-s + 1.04·11-s + 0.0418·13-s + 3.87·14-s − 0.0418·16-s + 4.75·17-s + 19-s + 1.07·20-s − 0.916·22-s + 6.87·23-s − 4.22·25-s − 0.0368·26-s + 5.41·28-s + 3.87·29-s + 7.78·31-s − 5.63·32-s − 4.18·34-s + 3.87·35-s + 1.22·37-s − 0.879·38-s − 2.49·40-s − 12.1·41-s + ⋯ |
| L(s) = 1 | − 0.621·2-s − 0.613·4-s − 0.393·5-s − 1.66·7-s + 1.00·8-s + 0.244·10-s + 0.314·11-s + 0.0116·13-s + 1.03·14-s − 0.0104·16-s + 1.15·17-s + 0.229·19-s + 0.241·20-s − 0.195·22-s + 1.43·23-s − 0.845·25-s − 0.00722·26-s + 1.02·28-s + 0.720·29-s + 1.39·31-s − 0.996·32-s − 0.717·34-s + 0.655·35-s + 0.201·37-s − 0.142·38-s − 0.394·40-s − 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6536732603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6536732603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 + 0.879T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.0418T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 + 1.14T + 97T^{2} \) |
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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
−10.50513503319045254161086110862, −9.884934418653644375321293915621, −9.241739815834792124005879287927, −8.358539468505201784556991737572, −7.36835470237737451780655245980, −6.47116922900830072076699954556, −5.27228976670993349300131871714, −3.97856978258585428568793670068, −3.05787287465741719831549846173, −0.813851737715328480645531095475,
0.813851737715328480645531095475, 3.05787287465741719831549846173, 3.97856978258585428568793670068, 5.27228976670993349300131871714, 6.47116922900830072076699954556, 7.36835470237737451780655245980, 8.358539468505201784556991737572, 9.241739815834792124005879287927, 9.884934418653644375321293915621, 10.50513503319045254161086110862
