| L(s) = 1 | − 2.26·2-s − 3-s + 3.11·4-s + 5-s + 2.26·6-s − 1.26·7-s − 2.52·8-s + 9-s − 2.26·10-s + 4.52·11-s − 3.11·12-s + 2.85·14-s − 15-s − 0.523·16-s − 4.49·17-s − 2.26·18-s + 5.11·19-s + 3.11·20-s + 1.26·21-s − 10.2·22-s + 2.23·23-s + 2.52·24-s + 25-s − 27-s − 3.93·28-s − 1.37·29-s + 2.26·30-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.447·5-s + 0.923·6-s − 0.476·7-s − 0.892·8-s + 0.333·9-s − 0.715·10-s + 1.36·11-s − 0.899·12-s + 0.762·14-s − 0.258·15-s − 0.130·16-s − 1.08·17-s − 0.533·18-s + 1.17·19-s + 0.696·20-s + 0.275·21-s − 2.18·22-s + 0.465·23-s + 0.515·24-s + 0.200·25-s − 0.192·27-s − 0.742·28-s − 0.255·29-s + 0.412·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7242890080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7242890080\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 + 0.231T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.52T + 53T^{2} \) |
| 59 | \( 1 + 0.853T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 9.60T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 6.62T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.059052696551416850029426786354, −8.360827733947179507965142659579, −7.38374377854596774315812180517, −6.61671312441268628995401131953, −6.34397721532183717445105786044, −5.12268696289500800055779870207, −4.08842349534311842685138679788, −2.79777535135361529284954269599, −1.62427717701669106561410556519, −0.74712520427779293049359071043,
0.74712520427779293049359071043, 1.62427717701669106561410556519, 2.79777535135361529284954269599, 4.08842349534311842685138679788, 5.12268696289500800055779870207, 6.34397721532183717445105786044, 6.61671312441268628995401131953, 7.38374377854596774315812180517, 8.360827733947179507965142659579, 9.059052696551416850029426786354
