| L(s) = 1 | + (1.36 + 0.366i)2-s + (−1.10 + 1.33i)3-s + (−2 + 1.41i)6-s + (−1.15 − 2.38i)7-s + (−1.99 − 2i)8-s + (−0.548 − 2.94i)9-s + (−3.67 − 2.12i)11-s + (−3.53 + 3.53i)13-s + (−0.707 − 3.67i)14-s + (−1.99 − 3.46i)16-s + (−0.366 − 1.36i)17-s + (0.330 − 4.22i)18-s + (6.06 − 3.5i)19-s + (4.44 + 1.09i)21-s + (−4.24 − 4.24i)22-s + (−1.09 + 4.09i)23-s + ⋯ |
| L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.639 + 0.769i)3-s + (−0.816 + 0.577i)6-s + (−0.436 − 0.899i)7-s + (−0.707 − 0.707i)8-s + (−0.182 − 0.983i)9-s + (−1.10 − 0.639i)11-s + (−0.980 + 0.980i)13-s + (−0.188 − 0.981i)14-s + (−0.499 − 0.866i)16-s + (−0.0887 − 0.331i)17-s + (0.0779 − 0.996i)18-s + (1.39 − 0.802i)19-s + (0.970 + 0.239i)21-s + (−0.904 − 0.904i)22-s + (−0.228 + 0.854i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.353474 - 0.497302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.353474 - 0.497302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.10 - 1.33i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.15 + 2.38i)T \) |
| good | 2 | \( 1 + (-1.36 - 0.366i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.53 - 3.53i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.366 + 1.36i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.06 + 3.5i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 - 4.09i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.258 - 0.965i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-3.53 + 3.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.46 + 1.46i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.73 - 0.732i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.965 - 0.258i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-1.81 - 6.76i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.06 + 3.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.77 - 13.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.07 + 7.07i)T + 97iT^{2} \) |
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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
−10.66733929883006699963134683489, −9.616485564868344406267749463238, −9.311140667028683235789172037807, −7.51284871392872494185612160519, −6.72737750837481212996482990116, −5.55432014172126909685212411777, −5.01495669645019096489598832163, −4.01247823140068807454590808698, −3.11443460388783436993011380845, −0.26701806996592273817600463919,
2.25331367513189857063684935753, 3.13096686300770899409732940220, 4.83319252247745026109728723870, 5.42407998357135991747609912164, 6.15717550888229688599973598784, 7.55207757486698462084230088526, 8.146639744584692384138684504663, 9.510939068669792585216539279931, 10.42090277215611840415519615823, 11.54845313063111359804898829897
