| L(s) = 1 | + 2-s + 3-s + 4-s − 2.85·5-s + 6-s + 3.66·7-s + 8-s + 9-s − 2.85·10-s + 6.42·11-s + 12-s + 3.41·13-s + 3.66·14-s − 2.85·15-s + 16-s + 2.55·17-s + 18-s − 19-s − 2.85·20-s + 3.66·21-s + 6.42·22-s − 4.12·23-s + 24-s + 3.13·25-s + 3.41·26-s + 27-s + 3.66·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.27·5-s + 0.408·6-s + 1.38·7-s + 0.353·8-s + 0.333·9-s − 0.901·10-s + 1.93·11-s + 0.288·12-s + 0.946·13-s + 0.979·14-s − 0.736·15-s + 0.250·16-s + 0.619·17-s + 0.235·18-s − 0.229·19-s − 0.637·20-s + 0.799·21-s + 1.37·22-s − 0.861·23-s + 0.204·24-s + 0.626·25-s + 0.668·26-s + 0.192·27-s + 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.700965446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.700965446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 0.101T + 31T^{2} \) |
| 37 | \( 1 + 0.395T + 37T^{2} \) |
| 41 | \( 1 + 0.901T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 0.0659T + 47T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 6.50T + 83T^{2} \) |
| 89 | \( 1 - 2.30T + 89T^{2} \) |
| 97 | \( 1 + 4.40T + 97T^{2} \) |
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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
−8.165210980581628752235271082163, −7.43379616921851322697844885797, −6.67636180468330075186351941846, −5.99969936279106983199158017384, −4.89041822144989055956119816777, −4.20724164863712606230360086912, −3.86452552800448662360044781352, −3.10912041244462860703215563357, −1.77107881204988379028351169743, −1.13810722007725777910731618674,
1.13810722007725777910731618674, 1.77107881204988379028351169743, 3.10912041244462860703215563357, 3.86452552800448662360044781352, 4.20724164863712606230360086912, 4.89041822144989055956119816777, 5.99969936279106983199158017384, 6.67636180468330075186351941846, 7.43379616921851322697844885797, 8.165210980581628752235271082163
