| L(s) = 1 | + 0.517i·2-s + 1.73·4-s + 3.31i·5-s + (−2.34 − 1.22i)7-s + 1.93i·8-s + 3·9-s − 1.71·10-s + 1.71·13-s + (0.633 − 1.21i)14-s + 2.46·16-s + 6.40·17-s + 1.55i·18-s − 4.69·19-s + 5.74i·20-s − 1.26·23-s + ⋯ |
| L(s) = 1 | + 0.366i·2-s + 0.866·4-s + 1.48i·5-s + (−0.886 − 0.462i)7-s + 0.683i·8-s + 9-s − 0.542·10-s + 0.476·13-s + (0.169 − 0.324i)14-s + 0.616·16-s + 1.55·17-s + 0.366i·18-s − 1.07·19-s + 1.28i·20-s − 0.264·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0681 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29471 + 1.38615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29471 + 1.38615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.34 + 1.22i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.517iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 3.31iT - 5T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 6.17iT - 29T^{2} \) |
| 31 | \( 1 - 9.06iT - 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 + 5.93iT - 43T^{2} \) |
| 47 | \( 1 + 9.06iT - 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 + 6.63iT - 59T^{2} \) |
| 61 | \( 1 + 8.12T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 7.46T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 - 9.14iT - 79T^{2} \) |
| 83 | \( 1 - 9.38T + 83T^{2} \) |
| 89 | \( 1 + 5.74iT - 89T^{2} \) |
| 97 | \( 1 + 8.17iT - 97T^{2} \) |
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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
−10.45897031916316022910243299924, −9.990375195633982492142665694070, −8.555907671241224241241751831976, −7.42704940963127015728407092952, −6.89381719352338761711576237904, −6.49418987982633936638246367620, −5.39849385359029354780089948305, −3.66417370630173297782238938181, −3.14339213341671896874831581936, −1.74505537690862354010693848202,
0.973306503229842216667901298336, 2.10477878581269082764527671632, 3.51384422451815620327906874267, 4.42098694950214484720806088234, 5.72330066818658061888507617418, 6.32018776207894156055798066611, 7.51470638696573923760359777225, 8.272502097578678288200024994054, 9.382633388420127550250720864339, 9.869598931861067217893529845285
