| L(s) = 1 | + (−0.948 − 1.16i)2-s + (1.18 − 1.02i)3-s + (−0.0542 + 0.259i)4-s + (0.841 + 2.65i)5-s + (−2.31 − 0.409i)6-s + (−1.01 + 2.95i)7-s + (−2.32 + 1.19i)8-s + (−0.0809 + 0.558i)9-s + (2.30 − 3.50i)10-s + (1.26 − 0.410i)11-s + (0.200 + 0.361i)12-s + (−0.0896 + 0.0302i)13-s + (4.42 − 1.61i)14-s + (3.70 + 2.27i)15-s + (4.08 + 1.78i)16-s + (2.91 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.670 − 0.825i)2-s + (0.681 − 0.589i)3-s + (−0.0271 + 0.129i)4-s + (0.376 + 1.18i)5-s + (−0.944 − 0.167i)6-s + (−0.384 + 1.11i)7-s + (−0.820 + 0.422i)8-s + (−0.0269 + 0.186i)9-s + (0.728 − 1.10i)10-s + (0.382 − 0.123i)11-s + (0.0578 + 0.104i)12-s + (−0.0248 + 0.00838i)13-s + (1.18 − 0.431i)14-s + (0.956 + 0.587i)15-s + (1.02 + 0.447i)16-s + (0.707 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21573 - 0.0268647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21573 - 0.0268647i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (-20.8 - 8.27i)T \) |
| good | 2 | \( 1 + (0.948 + 1.16i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-1.18 + 1.02i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 2.65i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (1.01 - 2.95i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (-1.26 + 0.410i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (0.0896 - 0.0302i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (-2.91 - 1.45i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-1.08 - 2.44i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.00 - 3.49i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-0.162 + 1.35i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (0.777 + 9.53i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (3.22 + 4.07i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (1.23 - 6.27i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (0.791 - 4.31i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (-4.81 + 0.180i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (0.962 - 2.15i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (-3.91 + 9.41i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (0.192 + 0.317i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (0.317 - 0.194i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (-0.0612 - 0.388i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (6.12 + 5.85i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (0.236 + 4.20i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (1.62 + 3.09i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (-1.82 - 6.95i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (1.85 + 7.87i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−10.85668905847640255849833664295, −9.924837990466535584900666886942, −9.363884566343067663411357416508, −8.385522194077252774333427088699, −7.52643880247198495292546854114, −6.26326093774069788387778475112, −5.64998358761174469565302201614, −3.37060035033556062330773839038, −2.57887381814771041386605409541, −1.76662566321054495570703127219,
0.875434540455532622731834556901, 3.17561787835157716350637095971, 4.14023211580194894880435852643, 5.32092146126718067028501809445, 6.65423060481390445140415495995, 7.34807661109354212196015847917, 8.612552722093634683989474356878, 8.857074347002566425080876774310, 9.763554524660990646137408401746, 10.36897798603712916819236521497
