| 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
−15.67134127522394359069999619074, −14.81793659898783593339042585673, −13.69251576145968721910593441587, −11.73098241149225277787898553239, −10.75424502862306388900421844112, −9.382431788926544066056594239037, −8.553734079840928067250670592314, −7.80779519353810313378075918278, −4.94193917820238455740438081653, −3.70290359740627271518352000857,
1.60528595067515800796628573997, 4.41665817608775769774915308159, 7.19438753214454412820785958906, 7.86906510341911814300996758653, 8.690621685136958948643866399905, 10.37856979954603710612061021004, 11.68315545608981178464192341730, 12.77293300624263959567467319616, 14.17914040264035791708899091783, 14.63677540860756159770553853814
