| L(s) = 1 | + (−4.37 + 6.01i)2-s + (−1.60 + 4.94i)3-s + (−12.1 − 37.3i)4-s + (−24.8 + 18.0i)5-s + (−22.7 − 31.2i)6-s + (40.6 − 13.1i)7-s + (164. + 53.5i)8-s + (−21.8 − 15.8i)9-s − 228. i·10-s + (−55.8 − 107. i)11-s + 204.·12-s + (−92.9 + 127. i)13-s + (−98.1 + 302. i)14-s + (−49.4 − 152. i)15-s + (−534. + 388. i)16-s + (−52.6 − 72.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 + 1.50i)2-s + (−0.178 + 0.549i)3-s + (−0.759 − 2.33i)4-s + (−0.995 + 0.723i)5-s + (−0.630 − 0.868i)6-s + (0.828 − 0.269i)7-s + (2.57 + 0.836i)8-s + (−0.269 − 0.195i)9-s − 2.28i·10-s + (−0.461 − 0.886i)11-s + 1.41·12-s + (−0.549 + 0.756i)13-s + (−0.500 + 1.54i)14-s + (−0.219 − 0.675i)15-s + (−2.08 + 1.51i)16-s + (−0.182 − 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0963 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0963 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0991556 - 0.0900251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0991556 - 0.0900251i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 - 4.94i)T \) |
| 11 | \( 1 + (55.8 + 107. i)T \) |
| good | 2 | \( 1 + (4.37 - 6.01i)T + (-4.94 - 15.2i)T^{2} \) |
| 5 | \( 1 + (24.8 - 18.0i)T + (193. - 594. i)T^{2} \) |
| 7 | \( 1 + (-40.6 + 13.1i)T + (1.94e3 - 1.41e3i)T^{2} \) |
| 13 | \( 1 + (92.9 - 127. i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (52.6 + 72.4i)T + (-2.58e4 + 7.94e4i)T^{2} \) |
| 19 | \( 1 + (624. + 202. i)T + (1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + 292.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (646. - 210. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-1.42e3 - 1.03e3i)T + (2.85e5 + 8.78e5i)T^{2} \) |
| 37 | \( 1 + (-56.5 - 173. i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 41 | \( 1 + (553. + 179. i)T + (2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 + 298. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (662. - 2.03e3i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (1.78e3 + 1.29e3i)T + (2.43e6 + 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-1.49e3 - 4.60e3i)T + (-9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + (1.17e3 + 1.62e3i)T + (-4.27e6 + 1.31e7i)T^{2} \) |
| 67 | \( 1 + 4.98e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-4.37e3 + 3.18e3i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-1.92e3 + 625. i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (1.82e3 - 2.50e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (3.44e3 + 4.73e3i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 + 1.48e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.29e3 - 3.84e3i)T + (2.73e7 + 8.41e7i)T^{2} \) |
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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
−16.76312293028636974613898721708, −15.76144645767398833489838477689, −14.96313021945279967504152570453, −14.08893762646893242749764179487, −11.31715221931061214775950620411, −10.43116786843935258322576197435, −8.778759391805505368492165322317, −7.76624946830354627190202645224, −6.50571415645607614060786698750, −4.67378024340112557519803007500,
0.13747566647020761975287223260, 2.05275430161302512057049190670, 4.39763184953176604380619893845, 7.85566083419686404893427603650, 8.368569476981832697628219030612, 10.10029353429548187414331657928, 11.36873419494683083393459674813, 12.27829715763966286900908492771, 12.93032587186319496426911088977, 15.18732680264862331779919050193
