| L(s) = 1 | + 1.29·2-s + 3-s − 0.331·4-s − 5-s + 1.29·6-s + 1.23·7-s − 3.01·8-s + 9-s − 1.29·10-s + 5.10·11-s − 0.331·12-s − 3.67·13-s + 1.59·14-s − 15-s − 3.22·16-s + 1.58·17-s + 1.29·18-s − 3.04·19-s + 0.331·20-s + 1.23·21-s + 6.59·22-s − 1.45·23-s − 3.01·24-s + 25-s − 4.74·26-s + 27-s − 0.407·28-s + ⋯ |
| L(s) = 1 | + 0.913·2-s + 0.577·3-s − 0.165·4-s − 0.447·5-s + 0.527·6-s + 0.465·7-s − 1.06·8-s + 0.333·9-s − 0.408·10-s + 1.53·11-s − 0.0955·12-s − 1.01·13-s + 0.425·14-s − 0.258·15-s − 0.807·16-s + 0.384·17-s + 0.304·18-s − 0.699·19-s + 0.0740·20-s + 0.268·21-s + 1.40·22-s − 0.302·23-s − 0.614·24-s + 0.200·25-s − 0.929·26-s + 0.192·27-s − 0.0770·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.376818874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.376818874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 5.10T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 - 9.92T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 59 | \( 1 - 0.106T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 7.97T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 + 3.39T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 - 3.58T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.246725432653366420417562440728, −7.25604319205688713473158054677, −6.62245892938443972298167715560, −5.89958736110677289380514205521, −4.84521938591794970908468827130, −4.44942478522674937681984870204, −3.78596591577546986534071545662, −3.02960586916753681331331239599, −2.11066297306885046304093763829, −0.831900023358599159753819112421,
0.831900023358599159753819112421, 2.11066297306885046304093763829, 3.02960586916753681331331239599, 3.78596591577546986534071545662, 4.44942478522674937681984870204, 4.84521938591794970908468827130, 5.89958736110677289380514205521, 6.62245892938443972298167715560, 7.25604319205688713473158054677, 8.246725432653366420417562440728
