| L(s) = 1 | + (−1.69 + 2.12i)2-s + (0.339 + 2.25i)3-s + (−1.20 − 5.26i)4-s + (−0.0789 − 0.115i)5-s + (−5.37 − 3.10i)6-s + (0.309 − 0.178i)7-s + (8.33 + 4.01i)8-s + (−2.10 + 0.648i)9-s + (0.380 + 0.0285i)10-s + (3.60 + 0.823i)11-s + (11.4 − 4.49i)12-s + (0.376 + 5.02i)13-s + (−0.144 + 0.959i)14-s + (0.234 − 0.217i)15-s + (−12.9 + 6.23i)16-s + (3.50 − 2.16i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 + 1.50i)2-s + (0.196 + 1.30i)3-s + (−0.600 − 2.63i)4-s + (−0.0353 − 0.0518i)5-s + (−2.19 − 1.26i)6-s + (0.116 − 0.0674i)7-s + (2.94 + 1.41i)8-s + (−0.700 + 0.216i)9-s + (0.120 + 0.00901i)10-s + (1.08 + 0.248i)11-s + (3.30 − 1.29i)12-s + (0.104 + 1.39i)13-s + (−0.0386 + 0.256i)14-s + (0.0605 − 0.0561i)15-s + (−3.23 + 1.55i)16-s + (0.851 − 0.524i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0106906 - 0.957038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0106906 - 0.957038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-3.50 + 2.16i)T \) |
| 43 | \( 1 + (-6.39 - 1.46i)T \) |
| good | 2 | \( 1 + (1.69 - 2.12i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.339 - 2.25i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (0.0789 + 0.115i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.178i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.60 - 0.823i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.376 - 5.02i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (-6.96 - 2.14i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-4.15 + 4.47i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.915 - 6.07i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (8.32 - 3.26i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (6.18 + 3.56i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.91 + 3.12i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (1.53 + 6.74i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.152 + 2.03i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.69 + 1.29i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-6.59 - 2.58i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-2.95 - 0.912i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (4.67 + 5.04i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-5.47 + 0.410i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (8.56 - 4.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.14 + 0.172i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (0.00618 - 0.000931i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (2.14 + 0.490i)T + (87.3 + 42.0i)T^{2} \) |
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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
−10.35834514748600165559089055465, −9.533043401068168701390792786311, −9.149886151675617926471413366593, −8.543029533256081737245764893467, −7.20349240181023228883850400151, −6.82238475274110504137403909573, −5.47417092633933782626602490241, −4.82477419273231628636366944196, −3.73109979589199853003226089269, −1.36646813424250826199088856850,
0.915206033141588868851269543757, 1.58087469406404409660973207591, 2.97067456285535649275901751803, 3.62283552861207842442768149878, 5.49421620588588928193715471253, 7.06092840787165394176731679529, 7.63172269841105291829591847401, 8.290079642066830288060295972165, 9.243065861359262413222641308268, 9.861771174936723899736438721721
