| L(s) = 1 | − 2-s − 0.406·3-s + 4-s − 5-s + 0.406·6-s − 7-s − 8-s − 2.83·9-s + 10-s + 11-s − 0.406·12-s − 0.813·13-s + 14-s + 0.406·15-s + 16-s + 3.83·17-s + 2.83·18-s + 5.42·19-s − 20-s + 0.406·21-s − 22-s − 5.42·23-s + 0.406·24-s + 25-s + 0.813·26-s + 2.37·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.234·3-s + 0.5·4-s − 0.447·5-s + 0.166·6-s − 0.377·7-s − 0.353·8-s − 0.944·9-s + 0.316·10-s + 0.301·11-s − 0.117·12-s − 0.225·13-s + 0.267·14-s + 0.105·15-s + 0.250·16-s + 0.930·17-s + 0.668·18-s + 1.24·19-s − 0.223·20-s + 0.0887·21-s − 0.213·22-s − 1.13·23-s + 0.0830·24-s + 0.200·25-s + 0.159·26-s + 0.456·27-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
| \(L(1)\) |
\(\approx\) |
\(0.8043523824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8043523824\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.406T + 3T^{2} \) |
| 13 | \( 1 + 0.813T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.978T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.20T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−10.05758193925894965327963395529, −9.672805827476162671586075452754, −8.437358141955009494890152436434, −7.988769677819698182606746893012, −6.89532147973069709931972235948, −6.05690711894951858180856458978, −5.08301946835727542382005756075, −3.62616469844236481638353617141, −2.64879365254247449832644212716, −0.828623097293271052336299898781,
0.828623097293271052336299898781, 2.64879365254247449832644212716, 3.62616469844236481638353617141, 5.08301946835727542382005756075, 6.05690711894951858180856458978, 6.89532147973069709931972235948, 7.988769677819698182606746893012, 8.437358141955009494890152436434, 9.672805827476162671586075452754, 10.05758193925894965327963395529
