L(s) = 1 | + 1.54i·2-s − 0.379·4-s + (−1.27 + 0.733i)5-s + (2.63 − 0.239i)7-s + 2.50i·8-s + (−1.13 − 1.95i)10-s + (1.93 − 1.11i)11-s + (3.57 + 0.453i)13-s + (0.369 + 4.06i)14-s − 4.61·16-s + 2.52·17-s + (−0.829 − 0.478i)19-s + (0.481 − 0.278i)20-s + (1.72 + 2.98i)22-s + 2.64·23-s + ⋯ |
L(s) = 1 | + 1.09i·2-s − 0.189·4-s + (−0.568 + 0.328i)5-s + (0.995 − 0.0904i)7-s + 0.883i·8-s + (−0.357 − 0.619i)10-s + (0.582 − 0.336i)11-s + (0.992 + 0.125i)13-s + (0.0986 + 1.08i)14-s − 1.15·16-s + 0.611·17-s + (−0.190 − 0.109i)19-s + (0.107 − 0.0622i)20-s + (0.367 + 0.635i)22-s + 0.552·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.971130 + 1.52632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.971130 + 1.52632i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.239i)T \) |
| 13 | \( 1 + (-3.57 - 0.453i)T \) |
good | 2 | \( 1 - 1.54iT - 2T^{2} \) |
| 5 | \( 1 + (1.27 - 0.733i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + (0.829 + 0.478i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + (-0.728 + 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.89 + 1.66i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.41iT - 37T^{2} \) |
| 41 | \( 1 + (3.52 + 2.03i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.05 - 5.22i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.74 - 3.02i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 0.767iT - 59T^{2} \) |
| 61 | \( 1 + (-6.05 + 10.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.35 - 4.82i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.50 + 1.44i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.3 - 6.56i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 + 3.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.89iT - 83T^{2} \) |
| 89 | \( 1 + 10.1iT - 89T^{2} \) |
| 97 | \( 1 + (5.44 - 3.14i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.74943669354843425777836269176, −9.378507594236337228684896076144, −8.377779816132336628459185819444, −7.961430313377927815921305080887, −7.06999227251347766788780743051, −6.28673226913051691194706571269, −5.39005686603149614552038950426, −4.37911862776241082416664477254, −3.21954897523759783798617142392, −1.54903786472318110317992296826,
1.04017859206059893673553574892, 2.05242494076380346570484274088, 3.49240601433490291683334641785, 4.17860085556575754408985395383, 5.28651002043570752357405389779, 6.53646505777631115588440022802, 7.52636611999618018089964616964, 8.437823174610453718112702966836, 9.180002495650992565298069876670, 10.25058440212111715797692023052