| L(s) = 1 | + 0.381·2-s + 3-s − 1.85·4-s + 0.381·6-s + 7-s − 1.47·8-s + 9-s − 1.85·12-s + 4.23·13-s + 0.381·14-s + 3.14·16-s + 7.85·17-s + 0.381·18-s − 0.854·19-s + 21-s + 4.23·23-s − 1.47·24-s + 1.61·26-s + 27-s − 1.85·28-s − 6·29-s + 5.09·31-s + 4.14·32-s + 3·34-s − 1.85·36-s + 1.76·37-s − 0.326·38-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 0.577·3-s − 0.927·4-s + 0.155·6-s + 0.377·7-s − 0.520·8-s + 0.333·9-s − 0.535·12-s + 1.17·13-s + 0.102·14-s + 0.786·16-s + 1.90·17-s + 0.0900·18-s − 0.195·19-s + 0.218·21-s + 0.883·23-s − 0.300·24-s + 0.317·26-s + 0.192·27-s − 0.350·28-s − 1.11·29-s + 0.914·31-s + 0.732·32-s + 0.514·34-s − 0.309·36-s + 0.289·37-s − 0.0529·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.909729708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.909729708\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + 1.14T + 97T^{2} \) |
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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
−8.080840489622750823121679918598, −7.12429803211799499034466894343, −6.27681289592562352492007188189, −5.45527806566659541069284961175, −5.04290135754926490914446504932, −4.05417553916341520000893590782, −3.54450732356475498764710643301, −2.92518860763549085071365741130, −1.60798906152915610546933895907, −0.841573588872889718975321529247,
0.841573588872889718975321529247, 1.60798906152915610546933895907, 2.92518860763549085071365741130, 3.54450732356475498764710643301, 4.05417553916341520000893590782, 5.04290135754926490914446504932, 5.45527806566659541069284961175, 6.27681289592562352492007188189, 7.12429803211799499034466894343, 8.080840489622750823121679918598
