| L(s) = 1 | − 2-s − 0.732·3-s + 4-s + 5-s + 0.732·6-s + 7-s − 8-s − 2.46·9-s − 10-s + 11-s − 0.732·12-s + 5.46·13-s − 14-s − 0.732·15-s + 16-s − 3.46·17-s + 2.46·18-s + 0.732·19-s + 20-s − 0.732·21-s − 22-s + 4.73·23-s + 0.732·24-s + 25-s − 5.46·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s + 0.447·5-s + 0.298·6-s + 0.377·7-s − 0.353·8-s − 0.821·9-s − 0.316·10-s + 0.301·11-s − 0.211·12-s + 1.51·13-s − 0.267·14-s − 0.189·15-s + 0.250·16-s − 0.840·17-s + 0.580·18-s + 0.167·19-s + 0.223·20-s − 0.159·21-s − 0.213·22-s + 0.986·23-s + 0.149·24-s + 0.200·25-s − 1.07·26-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
| \(L(1)\) |
\(\approx\) |
\(1.075210746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075210746\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.732T + 3T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−10.47237323446394615174819444828, −9.207637710995727186548757754648, −8.830300617531799884136822000500, −7.902350082028992676312604734823, −6.73693607303334863680108481089, −6.04624919771915157588960969122, −5.18252152488623258139706483363, −3.75480664012356713430521601096, −2.40419147338323849139663642914, −1.01564435355073207944759011600,
1.01564435355073207944759011600, 2.40419147338323849139663642914, 3.75480664012356713430521601096, 5.18252152488623258139706483363, 6.04624919771915157588960969122, 6.73693607303334863680108481089, 7.902350082028992676312604734823, 8.830300617531799884136822000500, 9.207637710995727186548757754648, 10.47237323446394615174819444828
