L(s) = 1 | + (−0.634 − 1.26i)2-s + (0.653 + 1.13i)3-s + (−1.19 + 1.60i)4-s + (−1.67 − 1.48i)5-s + (1.01 − 1.54i)6-s + (−1.03 − 2.43i)7-s + (2.78 + 0.494i)8-s + (0.646 − 1.11i)9-s + (−0.816 + 3.05i)10-s + (0.168 + 0.292i)11-s + (−2.59 − 0.305i)12-s − 1.29i·13-s + (−2.42 + 2.85i)14-s + (0.588 − 2.86i)15-s + (−1.14 − 3.83i)16-s + (−3.41 − 5.92i)17-s + ⋯ |
L(s) = 1 | + (−0.448 − 0.893i)2-s + (0.377 + 0.653i)3-s + (−0.597 + 0.801i)4-s + (−0.747 − 0.664i)5-s + (0.414 − 0.630i)6-s + (−0.391 − 0.920i)7-s + (0.984 + 0.174i)8-s + (0.215 − 0.373i)9-s + (−0.258 + 0.966i)10-s + (0.0508 + 0.0880i)11-s + (−0.749 − 0.0882i)12-s − 0.360i·13-s + (−0.646 + 0.762i)14-s + (0.151 − 0.739i)15-s + (−0.285 − 0.958i)16-s + (−0.829 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.357398 - 0.699794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.357398 - 0.699794i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.634 + 1.26i)T \) |
| 5 | \( 1 + (1.67 + 1.48i)T \) |
| 7 | \( 1 + (1.03 + 2.43i)T \) |
good | 3 | \( 1 + (-0.653 - 1.13i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.168 - 0.292i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.29iT - 13T^{2} \) |
| 17 | \( 1 + (3.41 + 5.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.551 + 0.318i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.51 + 2.62i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.90iT - 29T^{2} \) |
| 31 | \( 1 + (1.08 + 1.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.30 - 7.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.28iT - 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (-7.78 - 4.49i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.63 - 6.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.97 - 6.87i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 6.31i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.53iT - 71T^{2} \) |
| 73 | \( 1 + (5.93 + 10.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.96 - 5.17i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 + (6.43 + 3.71i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.03T + 97T^{2} \) |
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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
−11.43288805392297872348069825127, −10.52270278701028642844198148258, −9.595482029648103353992738380316, −8.994549503516406236619003625360, −7.926256634411307215929552892607, −6.94129223534653321954581400148, −4.71237194296814741273661362381, −4.08729583528079883313749171123, −2.97617827641454541661746330273, −0.67959540965697530984391193066,
2.06792561710204122888945975365, 3.86614547569631076281027975624, 5.42474686476888603163951606403, 6.65882816593688379071248770507, 7.20863262498588973764488296838, 8.376427922734397623801792239353, 8.842642939691440597926470275706, 10.25312848628530254975357145324, 11.07137037968824697521595011955, 12.38065156064061627389200021451