| L(s) = 1 | + 1.84·2-s − 3-s + 1.39·4-s − 5-s − 1.84·6-s + 0.480·7-s − 1.11·8-s + 9-s − 1.84·10-s − 4.58·11-s − 1.39·12-s + 5.01·13-s + 0.884·14-s + 15-s − 4.84·16-s + 5.00·17-s + 1.84·18-s − 0.0298·19-s − 1.39·20-s − 0.480·21-s − 8.44·22-s − 1.02·23-s + 1.11·24-s + 25-s + 9.23·26-s − 27-s + 0.669·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s − 0.577·3-s + 0.696·4-s − 0.447·5-s − 0.751·6-s + 0.181·7-s − 0.395·8-s + 0.333·9-s − 0.582·10-s − 1.38·11-s − 0.402·12-s + 1.39·13-s + 0.236·14-s + 0.258·15-s − 1.21·16-s + 1.21·17-s + 0.434·18-s − 0.00685·19-s − 0.311·20-s − 0.104·21-s − 1.79·22-s − 0.214·23-s + 0.228·24-s + 0.200·25-s + 1.81·26-s − 0.192·27-s + 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.511786743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511786743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 7 | \( 1 - 0.480T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + 0.0298T + 19T^{2} \) |
| 23 | \( 1 + 1.02T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 + 0.0874T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 + 1.51T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 3.60T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−8.000089569519685807847349538644, −7.19919360150580020967923128233, −6.36512990087192889206596291109, −5.70435289013952625500238270924, −5.22310891625036854542750771697, −4.57439240753335936333302418902, −3.63331478594411482897907301255, −3.24107373311754443545797145339, −2.06967241679581404930881945221, −0.69463811818563871880862393778,
0.69463811818563871880862393778, 2.06967241679581404930881945221, 3.24107373311754443545797145339, 3.63331478594411482897907301255, 4.57439240753335936333302418902, 5.22310891625036854542750771697, 5.70435289013952625500238270924, 6.36512990087192889206596291109, 7.19919360150580020967923128233, 8.000089569519685807847349538644
