| L(s) = 1 | + (−2.69 + 8.29i)2-s + (−7.28 + 5.29i)3-s + (−35.6 − 25.8i)4-s + (−23.0 − 70.9i)5-s + (−24.2 − 74.6i)6-s + (46.3 + 33.6i)7-s + (84.8 − 61.6i)8-s + (25.0 − 77.0i)9-s + 650.·10-s + (−338. + 215. i)11-s + 396.·12-s + (287. − 885. i)13-s + (−404. + 293. i)14-s + (543. + 394. i)15-s + (−152. − 470. i)16-s + (−303. − 935. i)17-s + ⋯ |
| L(s) = 1 | + (−0.476 + 1.46i)2-s + (−0.467 + 0.339i)3-s + (−1.11 − 0.808i)4-s + (−0.412 − 1.26i)5-s + (−0.274 − 0.846i)6-s + (0.357 + 0.259i)7-s + (0.468 − 0.340i)8-s + (0.103 − 0.317i)9-s + 2.05·10-s + (−0.843 + 0.537i)11-s + 0.794·12-s + (0.472 − 1.45i)13-s + (−0.551 + 0.400i)14-s + (0.623 + 0.452i)15-s + (−0.149 − 0.459i)16-s + (−0.255 − 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.268880 - 0.139634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.268880 - 0.139634i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.28 - 5.29i)T \) |
| 11 | \( 1 + (338. - 215. i)T \) |
| good | 2 | \( 1 + (2.69 - 8.29i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (23.0 + 70.9i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-46.3 - 33.6i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-287. + 885. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (303. + 935. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (365. - 265. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 4.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.84e3 + 1.34e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (400. - 1.23e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.59e3 - 1.88e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-2.44e3 + 1.77e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.46e4 - 1.06e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (3.26e3 - 1.00e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.21e4 - 2.33e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.25e4 - 3.85e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 4.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.22e4 + 6.85e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (5.78e4 + 4.20e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.49e4 - 4.60e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.46e4 + 7.58e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.33e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.80e3 + 2.09e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.86256303379400634199897595012, −14.95310651916901021848964166172, −13.18949230307181227846414639569, −11.89306853111079444113010738403, −10.01730362818112996997122361663, −8.559901057734478193285398413703, −7.73967566170128396214277192809, −5.77989017708391429492577170248, −4.80395800937877486385435540464, −0.21988669247997466389069506471,
2.03384750442295944882963919940, 3.81248901159474135846986119352, 6.51969620084758440085311977701, 8.219515706813907704844157934490, 10.08153749560363492641031371128, 11.09903987599400856464561746541, 11.57801587249280807440070749727, 13.09127035410269145333944209125, 14.38369960462676620521761356990, 16.02388030781206351620517767565
