| L(s) = 1 | + 2-s + 1.83·3-s + 4-s − 1.50·5-s + 1.83·6-s + 0.991·7-s + 8-s + 0.358·9-s − 1.50·10-s − 2.73·11-s + 1.83·12-s + 2.15·13-s + 0.991·14-s − 2.75·15-s + 16-s + 1.05·17-s + 0.358·18-s + 19-s − 1.50·20-s + 1.81·21-s − 2.73·22-s + 8.99·23-s + 1.83·24-s − 2.74·25-s + 2.15·26-s − 4.84·27-s + 0.991·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.672·5-s + 0.748·6-s + 0.374·7-s + 0.353·8-s + 0.119·9-s − 0.475·10-s − 0.826·11-s + 0.529·12-s + 0.597·13-s + 0.264·14-s − 0.711·15-s + 0.250·16-s + 0.256·17-s + 0.0844·18-s + 0.229·19-s − 0.336·20-s + 0.396·21-s − 0.584·22-s + 1.87·23-s + 0.374·24-s − 0.548·25-s + 0.422·26-s − 0.931·27-s + 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.459489812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.459489812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 - 0.991T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 23 | \( 1 - 8.99T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 - 2.18T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.67067991340533906086854375331, −7.51051017893340633594269842645, −6.42618710346335743747958330370, −5.59920380119104233934158662172, −4.97752975593242838411130121103, −4.09783529702493495618011727877, −3.50713330306040338269968646264, −2.82055160494025887447322166822, −2.15162385174781059386555505301, −0.896231035247059207479220374900,
0.896231035247059207479220374900, 2.15162385174781059386555505301, 2.82055160494025887447322166822, 3.50713330306040338269968646264, 4.09783529702493495618011727877, 4.97752975593242838411130121103, 5.59920380119104233934158662172, 6.42618710346335743747958330370, 7.51051017893340633594269842645, 7.67067991340533906086854375331
