L(s) = 1 | + (−0.531 + 1.98i)2-s + (−1.57 − 0.909i)3-s + (−1.91 − 1.10i)4-s + (2.75 + 0.737i)5-s + (2.64 − 2.64i)6-s + (0.311 − 0.311i)8-s + (0.155 + 0.269i)9-s + (−2.92 + 5.06i)10-s + (0.165 + 0.616i)11-s + (2.01 + 3.48i)12-s + (3.40 − 1.19i)13-s + (−3.66 − 3.66i)15-s + (−1.76 − 3.05i)16-s + (2.16 − 3.74i)17-s + (−0.616 + 0.165i)18-s + (4.64 + 1.24i)19-s + ⋯ |
L(s) = 1 | + (−0.375 + 1.40i)2-s + (−0.909 − 0.525i)3-s + (−0.958 − 0.553i)4-s + (1.23 + 0.329i)5-s + (1.07 − 1.07i)6-s + (0.109 − 0.109i)8-s + (0.0518 + 0.0898i)9-s + (−0.924 + 1.60i)10-s + (0.0498 + 0.186i)11-s + (0.581 + 1.00i)12-s + (0.943 − 0.330i)13-s + (−0.946 − 0.946i)15-s + (−0.440 − 0.763i)16-s + (0.524 − 0.908i)17-s + (−0.145 + 0.0389i)18-s + (1.06 + 0.285i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0201 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0201 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.797824 + 0.781885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.797824 + 0.781885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.40 + 1.19i)T \) |
good | 2 | \( 1 + (0.531 - 1.98i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 + (1.57 + 0.909i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.75 - 0.737i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.165 - 0.616i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 3.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.64 - 1.24i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.808 - 0.466i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + (-2.00 - 7.48i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.92 + 0.783i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.81 + 1.81i)T - 41iT^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (-2.16 + 8.07i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.68 - 2.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.349 + 0.0935i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (6.74 - 3.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.94 - 2.66i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-5.56 - 5.56i)T + 71iT^{2} \) |
| 73 | \( 1 + (-12.1 + 3.24i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.89 - 11.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.30 + 4.30i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.05 + 7.66i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.236 - 0.236i)T - 97iT^{2} \) |
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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
−10.62539258336121809503518983076, −9.738003551975661129401507602753, −8.968737830720353856232806846058, −7.919022487436442870330404127033, −6.94803821856912180813218039833, −6.39092616150274544248204688174, −5.67915661569107792046591069860, −5.10030216259398882181777524210, −2.99487735172443811442471023095, −1.14395953449459773342742429402,
1.01905074162542425071061284722, 2.17867533620045824118885712359, 3.53024693296831700462352935711, 4.70419740169491450233636249421, 5.80021082034667034706351028394, 6.33064693511059764452510429238, 8.142861238385380967228006327056, 9.157265021000377870835290916076, 9.774799175717311033336006014693, 10.48780281824213524967221471186