| 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
−11.01496569470377724733441389477, −10.10027527502706510589455400968, −9.449954801643310488373625264637, −8.655876145389535992539887372438, −7.20255871627372745366857635623, −6.73929327232907557934233644433, −5.16653170093605486096005007180, −4.52843288957643634468548620374, −3.36714081942324131810215801292, −1.71804996217205844677032267690,
0.69476271617776540840188386033, 2.68126242169535894628602431257, 3.89084938172619624247006206600, 4.73132594692689778678863262315, 6.11411668205387428187090837114, 7.38018868522824780122800954907, 7.941971853606460464081538254708, 8.751447145853820408504279919984, 9.337536318908286198272505104017, 10.95579337571986416827652232662
