| L(s) = 1 | + (−2.13 − 2.54i)2-s + (−1.22 + 6.94i)4-s + (2.35 + 6.46i)5-s + (1.10 + 6.28i)7-s + (8.77 − 5.06i)8-s + (11.4 − 19.8i)10-s + (−0.425 + 1.16i)11-s + (−4.09 − 3.43i)13-s + (13.6 − 16.2i)14-s + (−5.16 − 1.88i)16-s + (13.7 + 7.93i)17-s + (6.78 + 11.7i)19-s + (−47.7 + 8.42i)20-s + (3.88 − 1.41i)22-s + (−24.4 − 4.30i)23-s + ⋯ |
| L(s) = 1 | + (−1.06 − 1.27i)2-s + (−0.306 + 1.73i)4-s + (0.470 + 1.29i)5-s + (0.158 + 0.897i)7-s + (1.09 − 0.633i)8-s + (1.14 − 1.98i)10-s + (−0.0386 + 0.106i)11-s + (−0.314 − 0.264i)13-s + (0.973 − 1.16i)14-s + (−0.322 − 0.117i)16-s + (0.808 + 0.466i)17-s + (0.357 + 0.618i)19-s + (−2.38 + 0.421i)20-s + (0.176 − 0.0643i)22-s + (−1.06 − 0.187i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.735258 + 0.0565482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735258 + 0.0565482i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (2.13 + 2.54i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (-2.35 - 6.46i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 6.28i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (0.425 - 1.16i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (4.09 + 3.43i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-13.7 - 7.93i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.78 - 11.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (24.4 + 4.30i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-19.9 - 23.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.75 - 15.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-26.0 + 45.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-0.694 + 0.827i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (45.3 + 16.5i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-56.7 + 10.0i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 13.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (20.6 + 56.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-17.5 - 99.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (60.2 + 50.5i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (39.7 + 22.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (34.4 + 59.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-30.2 + 25.4i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-27.1 - 32.3i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-61.8 + 35.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (1.06 + 0.387i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−14.14509158724796577859795482709, −12.50563202923951965761728275675, −11.79097658491098803650491818513, −10.54362524932562277785556572878, −10.06354736706044426683489881629, −8.824115425235294768258072818441, −7.59085311161249315390532641126, −5.89149061125909904880466628145, −3.27372257324840895249886890248, −2.08869377102388558076614318646,
0.926018395617824736665470228361, 4.71902668802051542033956122367, 5.97199656860347294109009883466, 7.38534011943652877421505168661, 8.296651461865531663905306415396, 9.419504030674154416618632856479, 10.12509410397433276082718901937, 11.90081185816216806654109377447, 13.38052752340316281224387824141, 14.20917479322954490275493128778
