| L(s) = 1 | + (0.0158 + 0.00656i)2-s + (0.577 + 2.90i)3-s + (−2.82 − 2.82i)4-s + (7.34 + 4.90i)5-s + (−0.00990 + 0.0497i)6-s + (5.34 − 3.56i)7-s + (−0.0525 − 0.126i)8-s + (0.220 − 0.0913i)9-s + (0.0841 + 0.125i)10-s + (−3.69 − 0.735i)11-s + (6.57 − 9.84i)12-s + (9.53 − 9.53i)13-s + (0.108 − 0.0214i)14-s + (−10.0 + 24.1i)15-s + 15.9i·16-s + ⋯ |
| L(s) = 1 | + (0.00792 + 0.00328i)2-s + (0.192 + 0.967i)3-s + (−0.707 − 0.707i)4-s + (1.46 + 0.981i)5-s + (−0.00165 + 0.00829i)6-s + (0.762 − 0.509i)7-s + (−0.00656 − 0.0158i)8-s + (0.0244 − 0.0101i)9-s + (0.00841 + 0.0125i)10-s + (−0.336 − 0.0668i)11-s + (0.548 − 0.820i)12-s + (0.733 − 0.733i)13-s + (0.00771 − 0.00153i)14-s + (−0.666 + 1.60i)15-s + 0.999i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.94910 + 0.775538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94910 + 0.775538i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.0158 - 0.00656i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.577 - 2.90i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-7.34 - 4.90i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.34 + 3.56i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (3.69 + 0.735i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-9.53 + 9.53i)T - 169iT^{2} \) |
| 19 | \( 1 + (1.53 + 0.635i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (3.56 - 17.9i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-15.2 + 22.8i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (13.1 - 2.60i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-7.01 - 35.2i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-35.0 + 23.4i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (48.7 - 20.1i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (12.7 - 12.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (13.3 + 5.51i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (34.6 + 83.6i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (45.1 + 67.5i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 60.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (6.05 + 30.4i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (72.2 + 48.2i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-107. - 21.3i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-14.3 + 34.5i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-0.239 - 0.239i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (4.86 - 7.27i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−11.06986981562749624387800109235, −10.55782974304668088191225046837, −9.901788981854173760604012478162, −9.285039114592099499409696360353, −8.014183819374837931017084572800, −6.48047763094606935664243986389, −5.55906359035245746859314483414, −4.62968208223909562816041432326, −3.29111656640498082467396022073, −1.54531016487756525499108500983,
1.32366850726570511896170051042, 2.37140507741892200519922081197, 4.43770512168172975709548475516, 5.35986216611798753806660446534, 6.51223794074438199182453513836, 7.82909232801213028266340952784, 8.678885388583326699228327505005, 9.173455051991550886006275141883, 10.40834378061921072343566804870, 11.90299491658393386573292081999
