| L(s) = 1 | + 0.515·2-s + 3.41·3-s − 1.73·4-s − 5-s + 1.75·6-s − 3.94·7-s − 1.92·8-s + 8.64·9-s − 0.515·10-s − 5.91·12-s + 13-s − 2.03·14-s − 3.41·15-s + 2.47·16-s − 3.68·17-s + 4.45·18-s − 7.00·19-s + 1.73·20-s − 13.4·21-s − 0.500·23-s − 6.56·24-s + 25-s + 0.515·26-s + 19.2·27-s + 6.84·28-s + 1.44·29-s − 1.75·30-s + ⋯ |
| L(s) = 1 | + 0.364·2-s + 1.97·3-s − 0.867·4-s − 0.447·5-s + 0.717·6-s − 1.49·7-s − 0.680·8-s + 2.88·9-s − 0.162·10-s − 1.70·12-s + 0.277·13-s − 0.543·14-s − 0.881·15-s + 0.619·16-s − 0.894·17-s + 1.05·18-s − 1.60·19-s + 0.387·20-s − 2.93·21-s − 0.104·23-s − 1.34·24-s + 0.200·25-s + 0.101·26-s + 3.70·27-s + 1.29·28-s + 0.268·29-s − 0.321·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854021452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854021452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.515T + 2T^{2} \) |
| 3 | \( 1 - 3.41T + 3T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 0.500T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 - 4.11T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−8.110660795041338770631285698409, −7.18988378200382607556025473009, −6.65501759080188231121289179948, −5.84844098805112765409080810106, −4.48880614160035181877065674457, −4.10997821448445377441579563078, −3.60852198420034467568648831002, −2.81042084549721157150448217514, −2.25191285403935230556129024657, −0.70218081689062679014445902158,
0.70218081689062679014445902158, 2.25191285403935230556129024657, 2.81042084549721157150448217514, 3.60852198420034467568648831002, 4.10997821448445377441579563078, 4.48880614160035181877065674457, 5.84844098805112765409080810106, 6.65501759080188231121289179948, 7.18988378200382607556025473009, 8.110660795041338770631285698409
