| L(s) = 1 | + 2-s − 2.46·3-s + 4-s + 0.597·5-s − 2.46·6-s − 5.04·7-s + 8-s + 3.08·9-s + 0.597·10-s − 0.206·11-s − 2.46·12-s − 5.16·13-s − 5.04·14-s − 1.47·15-s + 16-s + 2.07·17-s + 3.08·18-s + 6.06·19-s + 0.597·20-s + 12.4·21-s − 0.206·22-s + 5.27·23-s − 2.46·24-s − 4.64·25-s − 5.16·26-s − 0.210·27-s − 5.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s + 0.267·5-s − 1.00·6-s − 1.90·7-s + 0.353·8-s + 1.02·9-s + 0.188·10-s − 0.0621·11-s − 0.712·12-s − 1.43·13-s − 1.34·14-s − 0.380·15-s + 0.250·16-s + 0.503·17-s + 0.727·18-s + 1.39·19-s + 0.133·20-s + 2.71·21-s − 0.0439·22-s + 1.10·23-s − 0.503·24-s − 0.928·25-s − 1.01·26-s − 0.0405·27-s − 0.952·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 0.597T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + 0.206T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 0.341T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 6.15T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 - 1.42T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 8.99T + 89T^{2} \) |
| 97 | \( 1 + 0.752T + 97T^{2} \) |
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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
−7.03990448301464529672545542534, −6.71630462892185476590811166241, −5.97798856130990739555838605766, −5.38699147696082679077440181994, −5.02063121510758301215425325043, −3.94934413643519587176074593924, −3.14755205193717877318971850249, −2.50523029392546985405182688077, −1.00809671949995590801904161175, 0,
1.00809671949995590801904161175, 2.50523029392546985405182688077, 3.14755205193717877318971850249, 3.94934413643519587176074593924, 5.02063121510758301215425325043, 5.38699147696082679077440181994, 5.97798856130990739555838605766, 6.71630462892185476590811166241, 7.03990448301464529672545542534
