| L(s) = 1 | + 0.0238·2-s − 1.99·4-s − 2.56·5-s + 0.894·7-s − 0.0954·8-s − 0.0611·10-s + 11-s − 2.49·13-s + 0.0213·14-s + 3.99·16-s + 2.00·17-s − 3.11·19-s + 5.12·20-s + 0.0238·22-s + 0.753·23-s + 1.57·25-s − 0.0596·26-s − 1.78·28-s − 8.63·29-s + 5.83·31-s + 0.286·32-s + 0.0478·34-s − 2.29·35-s − 3.01·37-s − 0.0743·38-s + 0.244·40-s + 4.20·41-s + ⋯ |
| L(s) = 1 | + 0.0168·2-s − 0.999·4-s − 1.14·5-s + 0.337·7-s − 0.0337·8-s − 0.0193·10-s + 0.301·11-s − 0.693·13-s + 0.00570·14-s + 0.999·16-s + 0.486·17-s − 0.714·19-s + 1.14·20-s + 0.00508·22-s + 0.157·23-s + 0.314·25-s − 0.0116·26-s − 0.337·28-s − 1.60·29-s + 1.04·31-s + 0.0506·32-s + 0.00820·34-s − 0.387·35-s − 0.496·37-s − 0.0120·38-s + 0.0386·40-s + 0.657·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7154068882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7154068882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0238T + 2T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 0.894T + 7T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 - 0.753T + 23T^{2} \) |
| 29 | \( 1 + 8.63T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 67 | \( 1 - 0.916T + 67T^{2} \) |
| 71 | \( 1 - 8.16T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 + 0.882T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.166444530172348476393489561960, −7.54367316369492033528469679303, −6.81286891848193840230367865837, −5.78931684107154316457788261153, −5.02913604120281180850459507642, −4.38175213296833127347457894404, −3.81252634583781490720881704715, −3.04097281382442112871823877644, −1.70746960615778958490186330395, −0.44878797025950631680857649509,
0.44878797025950631680857649509, 1.70746960615778958490186330395, 3.04097281382442112871823877644, 3.81252634583781490720881704715, 4.38175213296833127347457894404, 5.02913604120281180850459507642, 5.78931684107154316457788261153, 6.81286891848193840230367865837, 7.54367316369492033528469679303, 8.166444530172348476393489561960
