L(s) = 1 | + (−1.78 + 1.03i)2-s + (0.627 − 1.61i)3-s + (1.12 − 1.95i)4-s + (0.543 + 3.53i)6-s + (0.00953 + 2.64i)7-s + 0.527i·8-s + (−2.21 − 2.02i)9-s + (−4.06 − 2.34i)11-s + (−2.44 − 3.04i)12-s − 0.638i·13-s + (−2.74 − 4.71i)14-s + (1.71 + 2.96i)16-s + (2.07 − 3.59i)17-s + (6.04 + 1.33i)18-s + (−0.776 + 0.448i)19-s + ⋯ |
L(s) = 1 | + (−1.26 + 0.729i)2-s + (0.362 − 0.932i)3-s + (0.563 − 0.976i)4-s + (0.221 + 1.44i)6-s + (0.00360 + 0.999i)7-s + 0.186i·8-s + (−0.737 − 0.675i)9-s + (−1.22 − 0.707i)11-s + (−0.705 − 0.879i)12-s − 0.177i·13-s + (−0.733 − 1.26i)14-s + (0.427 + 0.741i)16-s + (0.503 − 0.871i)17-s + (1.42 + 0.315i)18-s + (−0.178 + 0.102i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.103235 - 0.241919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.103235 - 0.241919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.627 + 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.00953 - 2.64i)T \) |
good | 2 | \( 1 + (1.78 - 1.03i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (4.06 + 2.34i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.638iT - 13T^{2} \) |
| 17 | \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.89 - 3.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.14iT - 29T^{2} \) |
| 31 | \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.69 + 9.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 + 3.14T + 43T^{2} \) |
| 47 | \( 1 + (-3.40 - 5.89i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.96 + 1.13i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.22iT - 71T^{2} \) |
| 73 | \( 1 + (12.5 + 7.22i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−10.20370536117373487838457220876, −9.298589977428093008548140705578, −8.571101007441276391570917124691, −7.88278727748237048956218263092, −7.31077843728916635124952813444, −6.07348052205955950464084629214, −5.51760028758756860011316677244, −3.26455627585947485793209503392, −1.99738241235876270162158759337, −0.20988348753053853817253960257,
1.86641077128759345367090850158, 3.12321400451898660851628323824, 4.31129779718580540498510146153, 5.39655365288948191810993169691, 7.08444745785454169433691905131, 8.131809371297749188524393744242, 8.517973179198081254884837734156, 9.852695461572277812779755680406, 10.19872102228009055215904360561, 10.63622716142616278952139185389