L(s) = 1 | − 1.30·2-s + (−1.44 + 2.49i)3-s − 0.302·4-s + (−1.44 + 2.49i)5-s + (1.87 − 3.25i)6-s + 3·8-s + (−2.65 − 4.59i)9-s + (1.87 − 3.25i)10-s + (2.95 − 5.11i)11-s + (0.436 − 0.755i)12-s + (3.31 − 1.41i)13-s + (−4.15 − 7.19i)15-s − 3.30·16-s − 0.872·17-s + (3.45 + 5.98i)18-s + (−1.44 − 2.49i)19-s + ⋯ |
L(s) = 1 | − 0.921·2-s + (−0.831 + 1.44i)3-s − 0.151·4-s + (−0.644 + 1.11i)5-s + (0.766 − 1.32i)6-s + 1.06·8-s + (−0.883 − 1.53i)9-s + (0.593 − 1.02i)10-s + (0.890 − 1.54i)11-s + (0.125 − 0.218i)12-s + (0.920 − 0.391i)13-s + (−1.07 − 1.85i)15-s − 0.825·16-s − 0.211·17-s + (0.814 + 1.41i)18-s + (−0.330 − 0.572i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.466384 + 0.130024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.466384 + 0.130024i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.31 + 1.41i)T \) |
good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.44 - 2.49i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.95 + 5.11i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.872T + 17T^{2} \) |
| 19 | \( 1 + (1.44 + 2.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.436 + 0.755i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.19 + 10.7i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.80 + 8.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + (2.88 + 4.99i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.88 - 4.99i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.302 - 0.524i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + (-3.88 + 6.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.71523846634753745410436444068, −9.935898326429240579224384726168, −8.913519765712631460114741829223, −8.500928515868574203546895596723, −7.09669228964807819651706282036, −6.19374465385840783759406808372, −5.11242223815647225869610372543, −3.91763508375005680301039835934, −3.34516918117579994451855229286, −0.56274131094727279762342466635,
1.02926245911253534949341727189, 1.67215533806631131637059348646, 4.21614897193416035466260421634, 4.97190726669905956696375664226, 6.32812567060376394619412090400, 7.17892964567437183333514514018, 7.83397448089972946884450358315, 8.746605358177279788840623326297, 9.310774915865019746523057445773, 10.58098492257447477595484163011