| L(s) = 1 | − 2-s − 2.89·3-s + 4-s − 0.0804·5-s + 2.89·6-s + 2.75·7-s − 8-s + 5.38·9-s + 0.0804·10-s + 1.67·11-s − 2.89·12-s − 5.39·13-s − 2.75·14-s + 0.232·15-s + 16-s + 7.39·17-s − 5.38·18-s + 3.47·19-s − 0.0804·20-s − 7.97·21-s − 1.67·22-s + 0.325·23-s + 2.89·24-s − 4.99·25-s + 5.39·26-s − 6.91·27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.0359·5-s + 1.18·6-s + 1.04·7-s − 0.353·8-s + 1.79·9-s + 0.0254·10-s + 0.506·11-s − 0.835·12-s − 1.49·13-s − 0.735·14-s + 0.0601·15-s + 0.250·16-s + 1.79·17-s − 1.26·18-s + 0.796·19-s − 0.0179·20-s − 1.74·21-s − 0.357·22-s + 0.0678·23-s + 0.591·24-s − 0.998·25-s + 1.05·26-s − 1.32·27-s + 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8911570496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8911570496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 + 0.0804T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 7.39T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 - 0.325T + 23T^{2} \) |
| 29 | \( 1 - 9.31T + 29T^{2} \) |
| 31 | \( 1 - 0.749T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 - 7.03T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 16.8T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.224683341391862852697321530005, −7.62941298256145129657246093370, −7.08354527407557706977146688202, −6.20438707953782701853755695454, −5.45865589238312579819303278890, −4.97923733688251907710398039929, −4.12342950348715065593301040889, −2.72318227798098326645387691780, −1.43855316962853523015770400582, −0.73131950222491293984148192971,
0.73131950222491293984148192971, 1.43855316962853523015770400582, 2.72318227798098326645387691780, 4.12342950348715065593301040889, 4.97923733688251907710398039929, 5.45865589238312579819303278890, 6.20438707953782701853755695454, 7.08354527407557706977146688202, 7.62941298256145129657246093370, 8.224683341391862852697321530005
