| L(s) = 1 | + 2-s + 2.73·3-s + 4-s − 5-s + 2.73·6-s + 7-s + 8-s + 4.46·9-s − 10-s + 11-s + 2.73·12-s − 1.46·13-s + 14-s − 2.73·15-s + 16-s − 3.46·17-s + 4.46·18-s + 6.73·19-s − 20-s + 2.73·21-s + 22-s − 8.19·23-s + 2.73·24-s + 25-s − 1.46·26-s + 3.99·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.447·5-s + 1.11·6-s + 0.377·7-s + 0.353·8-s + 1.48·9-s − 0.316·10-s + 0.301·11-s + 0.788·12-s − 0.406·13-s + 0.267·14-s − 0.705·15-s + 0.250·16-s − 0.840·17-s + 1.05·18-s + 1.54·19-s − 0.223·20-s + 0.596·21-s + 0.213·22-s − 1.70·23-s + 0.557·24-s + 0.200·25-s − 0.287·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.708064299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.708064299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−10.16167403371029127339351710978, −9.378742497597221777073898978450, −8.503728533760204229827943501528, −7.69693598577128386953603209124, −7.15478314043648595030224249719, −5.80944923657199947454339965962, −4.53189308867694419899875178382, −3.79572434179813886888611896621, −2.84968810203567060751966738274, −1.80780264682836959243522545997,
1.80780264682836959243522545997, 2.84968810203567060751966738274, 3.79572434179813886888611896621, 4.53189308867694419899875178382, 5.80944923657199947454339965962, 7.15478314043648595030224249719, 7.69693598577128386953603209124, 8.503728533760204229827943501528, 9.378742497597221777073898978450, 10.16167403371029127339351710978
