L(s) = 1 | + (1.65 + 2.28i)2-s + (−0.535 − 1.64i)3-s + (−1.84 + 5.67i)4-s + (2.37 + 1.72i)5-s + (2.87 − 3.95i)6-s + (−10.6 + 3.45i)8-s + (−2.42 + 1.76i)9-s + 8.29i·10-s + (1.47 − 2.96i)11-s + 10.3·12-s + (2.11 + 2.91i)13-s + (1.57 − 4.84i)15-s + (−15.8 − 11.5i)16-s + (−8.05 − 2.61i)18-s + (−14.1 + 10.3i)20-s + ⋯ |
L(s) = 1 | + (1.17 + 1.61i)2-s + (−0.309 − 0.951i)3-s + (−0.921 + 2.83i)4-s + (1.06 + 0.772i)5-s + (1.17 − 1.61i)6-s + (−3.76 + 1.22i)8-s + (−0.809 + 0.587i)9-s + 2.62i·10-s + (0.445 − 0.895i)11-s + 2.98·12-s + (0.587 + 0.809i)13-s + (0.406 − 1.24i)15-s + (−3.97 − 2.88i)16-s + (−1.89 − 0.616i)18-s + (−3.17 + 2.30i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.814981 + 2.21213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.814981 + 2.21213i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.535 + 1.64i)T \) |
| 11 | \( 1 + (-1.47 + 2.96i)T \) |
| 13 | \( 1 + (-2.11 - 2.91i)T \) |
good | 2 | \( 1 + (-1.65 - 2.28i)T + (-0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.37 - 1.72i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-7.44 + 2.41i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.66iT - 43T^{2} \) |
| 47 | \( 1 + (0.508 + 1.56i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.92 + 12.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.35 - 1.86i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + (3.98 + 2.89i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-10.4 - 14.3i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.06 + 6.96i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + (-29.9 + 92.2i)T^{2} \) |
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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
−11.85279087656640103732039560115, −11.00102366065653787437304353186, −9.209044438303697285575447501534, −8.412613074297778212335268642240, −7.33712118567642876748416230207, −6.53752133104006734071227440308, −6.10037838599495201092621388887, −5.31829492685866923489682735624, −3.79601975320634885935706556635, −2.51648395049333028475951550414,
1.23058197322924602028789022355, 2.64707096153923871198020863511, 3.92250853414485477842193878236, 4.76637779148865449450913830149, 5.55462182419606995295455799498, 6.23662949389227937170360074064, 8.842680836490747725339885745816, 9.518022627216111929228456141679, 10.10657036188526758056022363612, 10.83997130702201984536601850818