| L(s) = 1 | + (−0.934 − 2.16i)2-s + (0.556 + 0.322i)3-s + (−2.45 + 2.59i)4-s + (−3.64 + 2.49i)5-s + (0.180 − 1.50i)6-s + (0.0510 − 0.299i)7-s + (3.49 + 1.27i)8-s + (−1.28 − 2.24i)9-s + (8.81 + 5.56i)10-s + (5.80 − 0.583i)11-s + (−2.20 + 0.653i)12-s + (−1.07 − 1.35i)13-s + (−0.696 + 0.168i)14-s + (−2.83 + 0.213i)15-s + (−0.0921 − 1.63i)16-s + (5.91 + 4.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.660 − 1.53i)2-s + (0.321 + 0.186i)3-s + (−1.22 + 1.29i)4-s + (−1.62 + 1.11i)5-s + (0.0735 − 0.615i)6-s + (0.0192 − 0.113i)7-s + (1.23 + 0.451i)8-s + (−0.427 − 0.748i)9-s + (2.78 + 1.76i)10-s + (1.75 − 0.175i)11-s + (−0.636 + 0.188i)12-s + (−0.297 − 0.375i)13-s + (−0.186 + 0.0451i)14-s + (−0.731 + 0.0550i)15-s + (−0.0230 − 0.408i)16-s + (1.43 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00786 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00786 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.565588 - 0.561158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.565588 - 0.561158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (-20.5 + 8.91i)T \) |
| good | 2 | \( 1 + (0.934 + 2.16i)T + (-1.37 + 1.45i)T^{2} \) |
| 3 | \( 1 + (-0.556 - 0.322i)T + (1.48 + 2.60i)T^{2} \) |
| 5 | \( 1 + (3.64 - 2.49i)T + (1.80 - 4.66i)T^{2} \) |
| 7 | \( 1 + (-0.0510 + 0.299i)T + (-6.60 - 2.32i)T^{2} \) |
| 11 | \( 1 + (-5.80 + 0.583i)T + (10.7 - 2.18i)T^{2} \) |
| 13 | \( 1 + (1.07 + 1.35i)T + (-2.98 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.91 - 4.50i)T + (4.51 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.46 + 2.35i)T + (-8.38 - 17.0i)T^{2} \) |
| 23 | \( 1 + (-1.26 + 0.753i)T + (10.9 - 20.2i)T^{2} \) |
| 29 | \( 1 + (1.59 - 0.139i)T + (28.5 - 5.05i)T^{2} \) |
| 31 | \( 1 + (5.44 + 1.24i)T + (27.9 + 13.5i)T^{2} \) |
| 37 | \( 1 + (-5.91 - 0.971i)T + (35.0 + 11.8i)T^{2} \) |
| 41 | \( 1 + (-8.84 - 1.68i)T + (38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (2.37 + 5.91i)T + (-31.0 + 29.7i)T^{2} \) |
| 47 | \( 1 + (3.62 - 0.502i)T + (45.2 - 12.7i)T^{2} \) |
| 53 | \( 1 + (-2.51 - 4.04i)T + (-23.3 + 47.5i)T^{2} \) |
| 59 | \( 1 + (1.47 - 0.185i)T + (57.1 - 14.6i)T^{2} \) |
| 61 | \( 1 + (-2.05 + 0.931i)T + (40.2 - 45.8i)T^{2} \) |
| 67 | \( 1 + (-7.15 - 0.538i)T + (66.2 + 10.0i)T^{2} \) |
| 71 | \( 1 + (-13.5 + 0.681i)T + (70.6 - 7.09i)T^{2} \) |
| 73 | \( 1 + (1.13 - 6.20i)T + (-68.2 - 25.9i)T^{2} \) |
| 79 | \( 1 + (1.13 + 5.43i)T + (-72.3 + 31.7i)T^{2} \) |
| 83 | \( 1 + (1.36 + 1.56i)T + (-10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (0.286 - 2.68i)T + (-86.9 - 18.7i)T^{2} \) |
| 97 | \( 1 + (3.30 - 1.11i)T + (77.1 - 58.7i)T^{2} \) |
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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
−10.91431815977393119453808224645, −9.946059134360195255913849304464, −9.117637963044615540504025657729, −8.278673941371980850121473176720, −7.41739364749441910638968559069, −6.26989282695600084632198407683, −3.93622623888925029230268257062, −3.71359221700529885738314170743, −2.80220473267709562895738137276, −0.837176261252790472463593159129,
1.00200793097586459283449535516, 3.63985725807641987684565302570, 4.77637628697717807588490586512, 5.59832217395603703521535348337, 7.04943026145347064820313034923, 7.61843463300963703423497083825, 8.218391563195087930608062396295, 9.104823850348932694632367402748, 9.519971119575976143650076204635, 11.38253249084668697346817823920
