| L(s) = 1 | + (15.6 − 27.1i)2-s + (119. + 206. i)3-s + (−234. − 406. i)4-s + (−908. − 1.57e3i)5-s + 7.48e3·6-s + (−4.89e3 − 8.48e3i)7-s + 1.34e3·8-s + (−1.86e4 + 3.23e4i)9-s − 5.68e4·10-s + 6.09e4·11-s + (5.60e4 − 9.70e4i)12-s + (−6.66e4 − 1.15e5i)13-s − 3.06e5·14-s + (2.16e5 − 3.75e5i)15-s + (1.41e5 − 2.44e5i)16-s + (1.30e5 − 2.25e5i)17-s + ⋯ |
| L(s) = 1 | + (0.692 − 1.19i)2-s + (0.851 + 1.47i)3-s + (−0.458 − 0.793i)4-s + (−0.649 − 1.12i)5-s + 2.35·6-s + (−0.770 − 1.33i)7-s + 0.115·8-s + (−0.949 + 1.64i)9-s − 1.79·10-s + 1.25·11-s + (0.779 − 1.35i)12-s + (−0.647 − 1.12i)13-s − 2.13·14-s + (1.10 − 1.91i)15-s + (0.538 − 0.932i)16-s + (0.377 − 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.46609 - 2.47629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46609 - 2.47629i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.13e7 - 1.42e6i)T \) |
| good | 2 | \( 1 + (-15.6 + 27.1i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-119. - 206. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (908. + 1.57e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (4.89e3 + 8.48e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 6.09e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (6.66e4 + 1.15e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.30e5 + 2.25e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.18e4 - 2.05e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + 5.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.49e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.89e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-1.57e7 - 2.72e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 2.67e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.62e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-2.33e7 + 4.03e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (6.73e7 - 1.16e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.50e7 - 2.60e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.03e8 + 1.78e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.82e8 - 3.15e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 3.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-7.56e7 - 1.31e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.74e8 + 4.74e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-3.99e6 + 6.92e6i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 7.48e8T + 7.60e17T^{2} \) |
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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
−13.81507611559596091868426863938, −12.79052552060906377717959395052, −11.50528758915905084131891977324, −10.15711667216065659097544424182, −9.501794462700691694535965987970, −7.86934034011280385766568171776, −4.77130800890854185863374157801, −4.00146859763060391922940800502, −3.15688511529993997655025555782, −0.78519183074930454753716004795,
2.08700868593479905346936753486, 3.59563531080504759441357548155, 6.21695929249005403982918140310, 6.76716144106696620577762016092, 7.80359049043224411128134897712, 9.154562424121996070306513485391, 11.75852461872118740491478130166, 12.63412574262687609444053005068, 14.00752991010669169720760751362, 14.65337496355059866611993580051
