| L(s) = 1 | + 2-s + 4-s − 2.89·5-s + 7-s + 8-s − 2.89·10-s + 11-s − 0.364·13-s + 14-s + 16-s − 2.89·17-s + 7.25·19-s − 2.89·20-s + 22-s + 8.14·23-s + 3.36·25-s − 0.364·26-s + 28-s − 2.36·29-s + 10.6·31-s + 32-s − 2.89·34-s − 2.89·35-s − 3.78·37-s + 7.25·38-s − 2.89·40-s + 2.89·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.29·5-s + 0.377·7-s + 0.353·8-s − 0.914·10-s + 0.301·11-s − 0.101·13-s + 0.267·14-s + 0.250·16-s − 0.701·17-s + 1.66·19-s − 0.646·20-s + 0.213·22-s + 1.69·23-s + 0.672·25-s − 0.0715·26-s + 0.188·28-s − 0.439·29-s + 1.91·31-s + 0.176·32-s − 0.496·34-s − 0.488·35-s − 0.622·37-s + 1.17·38-s − 0.457·40-s + 0.451·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.274720867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.274720867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2.89T + 5T^{2} \) |
| 13 | \( 1 + 0.364T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 - 8.14T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.52T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 3.10T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−9.500381757505576461744869066533, −8.680442375049077547121372089282, −7.68712141052455163926656827676, −7.27101098499960963634692509281, −6.29791579561805203836526474112, −5.12258704213171560060491740868, −4.50583889468581131997651863894, −3.58630503822586028230338006360, −2.72746456133807310065222655570, −1.03809843908470062371467227493,
1.03809843908470062371467227493, 2.72746456133807310065222655570, 3.58630503822586028230338006360, 4.50583889468581131997651863894, 5.12258704213171560060491740868, 6.29791579561805203836526474112, 7.27101098499960963634692509281, 7.68712141052455163926656827676, 8.680442375049077547121372089282, 9.500381757505576461744869066533
