| L(s) = 1 | + (−0.0144 + 0.0105i)2-s + (0.309 − 0.951i)3-s + (−0.617 + 1.90i)4-s + (−1.54 − 1.61i)5-s + (0.00552 + 0.0170i)6-s + 7-s + (−0.0221 − 0.0680i)8-s + (−0.809 − 0.587i)9-s + (0.0393 + 0.00718i)10-s + (3.99 − 2.90i)11-s + (1.61 + 1.17i)12-s + (−3.82 − 2.77i)13-s + (−0.0144 + 0.0105i)14-s + (−2.01 + 0.967i)15-s + (−3.23 − 2.35i)16-s + (0.930 + 2.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.0102 + 0.00743i)2-s + (0.178 − 0.549i)3-s + (−0.308 + 0.950i)4-s + (−0.690 − 0.723i)5-s + (0.00225 + 0.00694i)6-s + 0.377·7-s + (−0.00781 − 0.0240i)8-s + (−0.269 − 0.195i)9-s + (0.0124 + 0.00227i)10-s + (1.20 − 0.874i)11-s + (0.467 + 0.339i)12-s + (−1.06 − 0.770i)13-s + (−0.00386 + 0.00280i)14-s + (−0.520 + 0.249i)15-s + (−0.808 − 0.587i)16-s + (0.225 + 0.694i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0783 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0783 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.831916 - 0.769118i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.831916 - 0.769118i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (1.54 + 1.61i)T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + (0.0144 - 0.0105i)T + (0.618 - 1.90i)T^{2} \) |
| 11 | \( 1 + (-3.99 + 2.90i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.82 + 2.77i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.930 - 2.86i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.982 + 3.02i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.67 + 4.12i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.44 + 7.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.99 + 9.20i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.116 - 0.0845i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.23 + 3.08i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 + (1.95 - 6.02i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.42 - 10.5i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.03 - 2.20i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.93 - 6.48i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.695 - 2.13i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.63 + 11.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.10 + 2.25i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.10 - 9.55i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.37 + 4.24i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.70 + 1.24i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.87 - 18.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.95579337571986416827652232662, −9.337536318908286198272505104017, −8.751447145853820408504279919984, −7.941971853606460464081538254708, −7.38018868522824780122800954907, −6.11411668205387428187090837114, −4.73132594692689778678863262315, −3.89084938172619624247006206600, −2.68126242169535894628602431257, −0.69476271617776540840188386033,
1.71804996217205844677032267690, 3.36714081942324131810215801292, 4.52843288957643634468548620374, 5.16653170093605486096005007180, 6.73929327232907557934233644433, 7.20255871627372745366857635623, 8.655876145389535992539887372438, 9.449954801643310488373625264637, 10.10027527502706510589455400968, 11.01496569470377724733441389477
