| L(s) = 1 | − 0.165·2-s − 1.97·4-s − 4.22·5-s + 2.94·7-s + 0.657·8-s + 0.700·10-s − 4.55·11-s − 6.09·13-s − 0.487·14-s + 3.83·16-s + 4.56·17-s − 19-s + 8.34·20-s + 0.754·22-s − 7.05·23-s + 12.8·25-s + 1.00·26-s − 5.80·28-s + 0.586·29-s + 4.96·31-s − 1.95·32-s − 0.755·34-s − 12.4·35-s + 0.0505·37-s + 0.165·38-s − 2.78·40-s + 7.39·41-s + ⋯ |
| L(s) = 1 | − 0.117·2-s − 0.986·4-s − 1.89·5-s + 1.11·7-s + 0.232·8-s + 0.221·10-s − 1.37·11-s − 1.68·13-s − 0.130·14-s + 0.959·16-s + 1.10·17-s − 0.229·19-s + 1.86·20-s + 0.160·22-s − 1.47·23-s + 2.57·25-s + 0.197·26-s − 1.09·28-s + 0.108·29-s + 0.891·31-s − 0.344·32-s − 0.129·34-s − 2.10·35-s + 0.00830·37-s + 0.0268·38-s − 0.439·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.165T + 2T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 - 0.586T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 - 0.0505T + 37T^{2} \) |
| 41 | \( 1 - 7.39T + 41T^{2} \) |
| 43 | \( 1 - 9.02T + 43T^{2} \) |
| 53 | \( 1 - 5.05T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 - 9.71T + 71T^{2} \) |
| 73 | \( 1 + 6.79T + 73T^{2} \) |
| 79 | \( 1 + 8.34T + 79T^{2} \) |
| 83 | \( 1 + 3.89T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.61115809224873768097146738041, −7.43726508025791431347005249617, −5.89439363030052071064521880744, −5.05373800786695797814446884072, −4.63213114682045397447886922816, −4.14889379492503846830100668549, −3.21876950040350618304433231224, −2.34112962565924949985384201802, −0.833911794356724017051191186941, 0,
0.833911794356724017051191186941, 2.34112962565924949985384201802, 3.21876950040350618304433231224, 4.14889379492503846830100668549, 4.63213114682045397447886922816, 5.05373800786695797814446884072, 5.89439363030052071064521880744, 7.43726508025791431347005249617, 7.61115809224873768097146738041
