| L(s) = 1 | + 1.46·2-s + 3-s + 0.132·4-s + 5-s + 1.46·6-s + 1.86·7-s − 2.72·8-s + 9-s + 1.46·10-s − 0.482·11-s + 0.132·12-s + 3.37·13-s + 2.72·14-s + 15-s − 4.24·16-s + 4.48·17-s + 1.46·18-s − 0.154·19-s + 0.132·20-s + 1.86·21-s − 0.703·22-s + 6.12·23-s − 2.72·24-s + 25-s + 4.93·26-s + 27-s + 0.246·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.577·3-s + 0.0660·4-s + 0.447·5-s + 0.596·6-s + 0.705·7-s − 0.964·8-s + 0.333·9-s + 0.461·10-s − 0.145·11-s + 0.0381·12-s + 0.937·13-s + 0.728·14-s + 0.258·15-s − 1.06·16-s + 1.08·17-s + 0.344·18-s − 0.0353·19-s + 0.0295·20-s + 0.407·21-s − 0.150·22-s + 1.27·23-s − 0.556·24-s + 0.200·25-s + 0.967·26-s + 0.192·27-s + 0.0466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.592079487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.592079487\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 0.482T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 + 0.154T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 3.36T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.50T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 83 | \( 1 + 9.59T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−9.564033906793204907159647027968, −9.000854239026795104138843488683, −8.155788491579830381798325452771, −7.26852915026925718128876786066, −6.09965771884483118129715631355, −5.42871990076877164580759714069, −4.57334911500522165220864381997, −3.62274010986150048778075150646, −2.79610785972974282932119508305, −1.40825352935870174106497783548,
1.40825352935870174106497783548, 2.79610785972974282932119508305, 3.62274010986150048778075150646, 4.57334911500522165220864381997, 5.42871990076877164580759714069, 6.09965771884483118129715631355, 7.26852915026925718128876786066, 8.155788491579830381798325452771, 9.000854239026795104138843488683, 9.564033906793204907159647027968
