| L(s) = 1 | + 2.15·2-s − 0.661·3-s + 2.64·4-s + 5-s − 1.42·6-s + 1.08·7-s + 1.37·8-s − 2.56·9-s + 2.15·10-s − 0.580·11-s − 1.74·12-s − 2.47·13-s + 2.33·14-s − 0.661·15-s − 2.30·16-s + 17-s − 5.51·18-s + 6.07·19-s + 2.64·20-s − 0.715·21-s − 1.24·22-s − 7.59·23-s − 0.913·24-s + 25-s − 5.32·26-s + 3.68·27-s + 2.85·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s − 0.382·3-s + 1.32·4-s + 0.447·5-s − 0.582·6-s + 0.408·7-s + 0.487·8-s − 0.853·9-s + 0.681·10-s − 0.174·11-s − 0.504·12-s − 0.686·13-s + 0.622·14-s − 0.170·15-s − 0.577·16-s + 0.242·17-s − 1.30·18-s + 1.39·19-s + 0.590·20-s − 0.156·21-s − 0.266·22-s − 1.58·23-s − 0.186·24-s + 0.200·25-s − 1.04·26-s + 0.708·27-s + 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.661T + 3T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 + 0.580T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 19 | \( 1 - 6.07T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 + 9.83T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 - 1.98T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 - 0.758T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 + 1.91T + 79T^{2} \) |
| 83 | \( 1 - 5.78T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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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
−7.57492343052386018855145047267, −6.71111286042114971121009371483, −5.96064598739813378602136533567, −5.29750173479401466976891507829, −5.21843980817681657954146855030, −4.08291060500774693806735459112, −3.38794925434379034981296551498, −2.54850821380160014119651713946, −1.73625170121692851294681042524, 0,
1.73625170121692851294681042524, 2.54850821380160014119651713946, 3.38794925434379034981296551498, 4.08291060500774693806735459112, 5.21843980817681657954146855030, 5.29750173479401466976891507829, 5.96064598739813378602136533567, 6.71111286042114971121009371483, 7.57492343052386018855145047267
