| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 5-s − 7-s − 2.23·8-s − 1.61·10-s − 4.23·11-s + 1.38·13-s − 1.61·14-s − 4.85·16-s − 1.61·17-s − 7.09·19-s − 0.618·20-s − 6.85·22-s − 5.38·23-s + 25-s + 2.23·26-s − 0.618·28-s + 9.56·29-s + 6.70·31-s − 3.38·32-s − 2.61·34-s + 35-s + 6.70·37-s − 11.4·38-s + 2.23·40-s − 8.09·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.447·5-s − 0.377·7-s − 0.790·8-s − 0.511·10-s − 1.27·11-s + 0.383·13-s − 0.432·14-s − 1.21·16-s − 0.392·17-s − 1.62·19-s − 0.138·20-s − 1.46·22-s − 1.12·23-s + 0.200·25-s + 0.438·26-s − 0.116·28-s + 1.77·29-s + 1.20·31-s − 0.597·32-s − 0.448·34-s + 0.169·35-s + 1.10·37-s − 1.86·38-s + 0.353·40-s − 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 9.56T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 + 11T + 47T^{2} \) |
| 53 | \( 1 - 4.38T + 53T^{2} \) |
| 59 | \( 1 + 1.29T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 1.47T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−9.834814892749257393832066142655, −8.455068163802194315339698757933, −8.177582226516844944927988643487, −6.62652671638674137004661111524, −6.19647478656595924400803890664, −4.96678938605535477894573788617, −4.38999148689490299031447079082, −3.33569908990296030142128982539, −2.40997672140956335910460485092, 0,
2.40997672140956335910460485092, 3.33569908990296030142128982539, 4.38999148689490299031447079082, 4.96678938605535477894573788617, 6.19647478656595924400803890664, 6.62652671638674137004661111524, 8.177582226516844944927988643487, 8.455068163802194315339698757933, 9.834814892749257393832066142655
