| L(s) = 1 | + (−1.22 − 0.193i)2-s + (1.15 − 2.26i)3-s + (−0.443 − 0.144i)4-s + (−2.23 + 0.160i)5-s + (−1.85 + 2.54i)6-s + (3.09 − 1.57i)7-s + (2.72 + 1.38i)8-s + (−2.04 − 2.80i)9-s + (2.75 + 0.235i)10-s + (0.937 + 3.18i)11-s + (−0.839 + 0.839i)12-s + (−0.115 + 0.730i)13-s + (−4.08 + 1.32i)14-s + (−2.21 + 5.24i)15-s + (−2.30 − 1.67i)16-s + (0.0965 + 0.609i)17-s + ⋯ |
| L(s) = 1 | + (−0.864 − 0.136i)2-s + (0.666 − 1.30i)3-s + (−0.221 − 0.0720i)4-s + (−0.997 + 0.0718i)5-s + (−0.755 + 1.04i)6-s + (1.16 − 0.595i)7-s + (0.962 + 0.490i)8-s + (−0.680 − 0.936i)9-s + (0.872 + 0.0744i)10-s + (0.282 + 0.959i)11-s + (−0.242 + 0.242i)12-s + (−0.0320 + 0.202i)13-s + (−1.09 + 0.354i)14-s + (−0.571 + 1.35i)15-s + (−0.576 − 0.418i)16-s + (0.0234 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.473513 - 0.391718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.473513 - 0.391718i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 - 0.160i)T \) |
| 11 | \( 1 + (-0.937 - 3.18i)T \) |
| good | 2 | \( 1 + (1.22 + 0.193i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-1.15 + 2.26i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (-3.09 + 1.57i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (0.115 - 0.730i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.0965 - 0.609i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.971 - 2.99i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.30 + 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.896 - 2.75i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.45 - 1.78i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.04 + 2.04i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.970 + 0.315i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.07 - 4.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.967 - 0.492i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (4.24 + 0.671i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-7.03 - 2.28i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.20 - 3.03i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (9.39 - 9.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.92 + 2.12i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.244 + 0.479i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (9.87 - 7.17i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-15.8 + 2.50i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (-2.24 + 14.1i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−14.63677540860756159770553853814, −14.17914040264035791708899091783, −12.77293300624263959567467319616, −11.68315545608981178464192341730, −10.37856979954603710612061021004, −8.690621685136958948643866399905, −7.86906510341911814300996758653, −7.19438753214454412820785958906, −4.41665817608775769774915308159, −1.60528595067515800796628573997,
3.70290359740627271518352000857, 4.94193917820238455740438081653, 7.80779519353810313378075918278, 8.553734079840928067250670592314, 9.382431788926544066056594239037, 10.75424502862306388900421844112, 11.73098241149225277787898553239, 13.69251576145968721910593441587, 14.81793659898783593339042585673, 15.67134127522394359069999619074
