L(s) = 1 | + (−1.07 − 0.921i)2-s + (0.300 + 1.97i)4-s + (0.316 − 0.548i)5-s + (−2.06 + 1.65i)7-s + (1.50 − 2.39i)8-s + (−0.846 + 0.296i)10-s + (−0.424 + 0.245i)11-s + 3.13i·13-s + (3.73 + 0.125i)14-s + (−3.81 + 1.18i)16-s + (0.987 − 0.569i)17-s + (−0.591 + 1.02i)19-s + (1.18 + 0.462i)20-s + (0.681 + 0.128i)22-s + (−2.80 + 4.85i)23-s + ⋯ |
L(s) = 1 | + (−0.758 − 0.651i)2-s + (0.150 + 0.988i)4-s + (0.141 − 0.245i)5-s + (−0.779 + 0.626i)7-s + (0.530 − 0.847i)8-s + (−0.267 + 0.0937i)10-s + (−0.127 + 0.0738i)11-s + 0.868i·13-s + (0.999 + 0.0335i)14-s + (−0.954 + 0.296i)16-s + (0.239 − 0.138i)17-s + (−0.135 + 0.234i)19-s + (0.263 + 0.103i)20-s + (0.145 + 0.0274i)22-s + (−0.584 + 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.553815 + 0.353311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.553815 + 0.353311i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.07 + 0.921i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.06 - 1.65i)T \) |
good | 5 | \( 1 + (-0.316 + 0.548i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.424 - 0.245i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 + (-0.987 + 0.569i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.591 - 1.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.80 - 4.85i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 + (5.40 - 3.11i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.53 - 3.77i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.06iT - 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 + (4.80 - 8.32i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.10 + 1.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.12 + 5.26i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.1 - 5.87i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 - 5.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + (5.35 + 9.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.49 + 1.43i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 + (9.12 + 5.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−11.17978534119742161122715189631, −9.908690793380419672676551871610, −9.451916642552801324238618179900, −8.690005423373328209281600400023, −7.62359028111374261790840093419, −6.67967575671355547032341224322, −5.50933784761315457931460583525, −4.03851175391800454928554707527, −2.93972976309651091295391212859, −1.64164699762363402287594392135,
0.49978545201178609805242524179, 2.46253128521763371045147599136, 4.00781113006580780882194683432, 5.45385388904715388425738798297, 6.30148018960665050690019428355, 7.16614208503961678030036612872, 7.990258272040460011007337766551, 8.952221428620927159148024477071, 9.949101212843335909283667426832, 10.43715789290578220850549287778