| L(s) = 1 | + (0.632 + 0.856i)2-s + (0.641 + 0.121i)3-s + (0.255 − 0.827i)4-s + (0.264 + 2.22i)5-s + (0.301 + 0.626i)6-s + (0.776 + 2.52i)7-s + (2.88 − 1.00i)8-s + (−2.39 − 0.940i)9-s + (−1.73 + 1.63i)10-s + (−0.470 − 1.19i)11-s + (0.264 − 0.499i)12-s + (1.41 − 0.159i)13-s + (−1.67 + 2.26i)14-s + (−0.0996 + 1.45i)15-s + (1.25 + 0.855i)16-s + (1.42 − 0.0531i)17-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.605i)2-s + (0.370 + 0.0700i)3-s + (0.127 − 0.413i)4-s + (0.118 + 0.992i)5-s + (0.123 + 0.255i)6-s + (0.293 + 0.955i)7-s + (1.01 − 0.356i)8-s + (−0.798 − 0.313i)9-s + (−0.548 + 0.515i)10-s + (−0.141 − 0.361i)11-s + (0.0762 − 0.144i)12-s + (0.393 − 0.0442i)13-s + (−0.447 + 0.605i)14-s + (−0.0257 + 0.375i)15-s + (0.313 + 0.213i)16-s + (0.344 − 0.0128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61686 + 0.873102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61686 + 0.873102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.264 - 2.22i)T \) |
| 7 | \( 1 + (-0.776 - 2.52i)T \) |
| good | 2 | \( 1 + (-0.632 - 0.856i)T + (-0.589 + 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.641 - 0.121i)T + (2.79 + 1.09i)T^{2} \) |
| 11 | \( 1 + (0.470 + 1.19i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.41 + 0.159i)T + (12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 0.0531i)T + (16.9 - 1.27i)T^{2} \) |
| 19 | \( 1 + (-0.606 - 1.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0807 + 2.15i)T + (-22.9 - 1.71i)T^{2} \) |
| 29 | \( 1 + (-2.45 - 0.561i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (5.39 + 3.11i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (10.5 + 5.58i)T + (20.8 + 30.5i)T^{2} \) |
| 41 | \( 1 + (-1.62 + 3.37i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (1.51 - 4.34i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (-8.75 + 6.45i)T + (13.8 - 44.9i)T^{2} \) |
| 53 | \( 1 + (-2.80 + 1.48i)T + (29.8 - 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.0385 + 0.514i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.350 - 1.13i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (6.04 - 1.61i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (1.52 + 6.68i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 8.20i)T + (21.5 + 69.7i)T^{2} \) |
| 79 | \( 1 + (7.23 - 4.17i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.967 + 8.58i)T + (-80.9 - 18.4i)T^{2} \) |
| 89 | \( 1 + (6.50 - 16.5i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (3.80 - 3.80i)T - 97iT^{2} \) |
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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
−12.22176547319371306146539288526, −11.20063821074624473222163522949, −10.45440803754820883259631388769, −9.265386397634394012874217406179, −8.240429563368053110372439308141, −7.07915875788693988054591560311, −5.98744336888476334126141377876, −5.43288928490705760822984816000, −3.62916629563698320948485266031, −2.28541108092510259824693150085,
1.66592865167433396640112973006, 3.25456614015892708785589805032, 4.38830760666721234551788234057, 5.39836737244402720624264028996, 7.20752920430710663006389727815, 8.079236410601244155696419072634, 8.893288248732177377613222779026, 10.24551578207812100360534269969, 11.16132982567717009769844330642, 12.03288690312925934249465663270
