| L(s) = 1 | + (0.366 + 1.36i)2-s + (0.957 + 3.57i)3-s + (−1.73 + i)4-s + (4.99 − 0.300i)5-s + (−4.52 + 2.61i)6-s + (9.12 + 2.44i)7-s + (−2 − 1.99i)8-s + (−4.04 + 2.33i)9-s + (2.23 + 6.70i)10-s + (−15.1 − 8.74i)11-s + (−5.22 − 5.22i)12-s + (−6.39 + 11.3i)13-s + 13.3i·14-s + (5.85 + 17.5i)15-s + (1.99 − 3.46i)16-s + (1.68 − 6.27i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.683i)2-s + (0.319 + 1.19i)3-s + (−0.433 + 0.250i)4-s + (0.998 − 0.0601i)5-s + (−0.754 + 0.435i)6-s + (1.30 + 0.349i)7-s + (−0.250 − 0.249i)8-s + (−0.449 + 0.259i)9-s + (0.223 + 0.670i)10-s + (−1.37 − 0.795i)11-s + (−0.435 − 0.435i)12-s + (−0.491 + 0.870i)13-s + 0.954i·14-s + (0.390 + 1.16i)15-s + (0.124 − 0.216i)16-s + (0.0989 − 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08362 + 1.58952i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08362 + 1.58952i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 5 | \( 1 + (-4.99 + 0.300i)T \) |
| 13 | \( 1 + (6.39 - 11.3i)T \) |
| good | 3 | \( 1 + (-0.957 - 3.57i)T + (-7.79 + 4.5i)T^{2} \) |
| 7 | \( 1 + (-9.12 - 2.44i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (15.1 + 8.74i)T + (60.5 + 104. i)T^{2} \) |
| 17 | \( 1 + (-1.68 + 6.27i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (16.0 + 27.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (6.59 + 24.6i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-43.3 - 25.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 31.1iT - 961T^{2} \) |
| 37 | \( 1 + (2.27 + 8.50i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-3.28 - 1.89i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (40.7 + 10.9i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-34.7 + 34.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (26.8 - 26.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (32.8 + 56.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.4 + 38.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.6 - 43.5i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-29.4 + 17.0i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-6.68 - 6.68i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (29.6 + 29.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-5.93 + 10.2i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-40.0 - 10.7i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−13.88541141212203776973256981949, −12.58012714110165498320037278118, −11.00333580514984300608151456110, −10.23918059295592119441473056160, −8.959092978850381493321391997043, −8.417547799960797786280670864059, −6.72232435085316746055729602122, −5.09096416219998150132989338929, −4.76543363410302259569638432893, −2.63938350249961095528946364449,
1.55175565812472029907987918625, 2.45892792081836345899663808527, 4.74950477434434299990862090177, 5.95538196439154870560878454972, 7.66964776963599867621572035323, 8.154860056643869614462042125774, 10.01104687295163898920435694832, 10.55284294773478768114389518912, 12.03615094098272617770613564802, 12.85532647786418071963437869670
