| L(s) = 1 | + (−0.422 − 1.16i)2-s + (−1.51 + 0.846i)3-s + (0.362 − 0.304i)4-s + (0.253 + 1.44i)5-s + (1.62 + 1.39i)6-s + (1.07 − 2.41i)7-s + (−2.64 − 1.52i)8-s + (1.56 − 2.55i)9-s + (1.56 − 0.903i)10-s + (4.32 + 0.762i)11-s + (−0.290 + 0.767i)12-s + (1.72 − 4.72i)13-s + (−3.26 − 0.226i)14-s + (−1.60 − 1.96i)15-s + (−0.491 + 2.78i)16-s + (0.691 + 1.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.298 − 0.820i)2-s + (−0.872 + 0.488i)3-s + (0.181 − 0.152i)4-s + (0.113 + 0.644i)5-s + (0.661 + 0.570i)6-s + (0.406 − 0.913i)7-s + (−0.935 − 0.540i)8-s + (0.521 − 0.852i)9-s + (0.494 − 0.285i)10-s + (1.30 + 0.229i)11-s + (−0.0838 + 0.221i)12-s + (0.477 − 1.31i)13-s + (−0.871 − 0.0604i)14-s + (−0.414 − 0.506i)15-s + (−0.122 + 0.696i)16-s + (0.167 + 0.290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.697716 - 0.580299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.697716 - 0.580299i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.51 - 0.846i)T \) |
| 7 | \( 1 + (-1.07 + 2.41i)T \) |
| good | 2 | \( 1 + (0.422 + 1.16i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.253 - 1.44i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-4.32 - 0.762i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.72 + 4.72i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.691 - 1.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.13 + 2.38i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.82 + 4.55i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.96 - 5.40i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.06 - 4.84i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.26 - 0.459i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.613 - 3.47i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.55 + 4.66i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 0.811iT - 53T^{2} \) |
| 59 | \( 1 + (0.933 + 5.29i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (5.13 - 6.11i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.21 + 1.53i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (11.6 - 6.74i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.23 - 3.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.07 - 2.93i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.57 + 1.66i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.219 + 0.380i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.66 + 0.646i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.00352476773724188174155854464, −11.11632599706619289583021266884, −10.46108422690735615730952131038, −10.02634624000172482311276262173, −8.580197506344481043378483356288, −6.78770384764984003442809454482, −6.24801267197001530398843155174, −4.53928877241373325188951616923, −3.25467490076384856586476694692, −1.13243578284592558176413539513,
1.85870710959174094818170133184, 4.38082456540012956913161217983, 5.91981274221561037350602108065, 6.30771890185930967717203968717, 7.60843768148718240021335798342, 8.631216386750205729093786713768, 9.420938912490116339879774984297, 11.29998681146432341324321721218, 11.79130110570399022163381868210, 12.46865908960282850055190362055
