| L(s) = 1 | + (2.31 − 0.408i)2-s + (1.43 − 0.523i)4-s + (3.71 − 4.42i)5-s + (4.57 + 1.66i)7-s + (−5.02 + 2.90i)8-s + (6.79 − 11.7i)10-s + (−1.90 − 2.27i)11-s + (−3.38 + 19.1i)13-s + (11.2 + 1.98i)14-s + (−15.1 + 12.7i)16-s + (−21.7 − 12.5i)17-s + (−8.92 − 15.4i)19-s + (3.02 − 8.31i)20-s + (−5.34 − 4.48i)22-s + (6.02 + 16.5i)23-s + ⋯ |
| L(s) = 1 | + (1.15 − 0.204i)2-s + (0.359 − 0.130i)4-s + (0.743 − 0.885i)5-s + (0.653 + 0.237i)7-s + (−0.628 + 0.362i)8-s + (0.679 − 1.17i)10-s + (−0.173 − 0.206i)11-s + (−0.260 + 1.47i)13-s + (0.805 + 0.142i)14-s + (−0.947 + 0.794i)16-s + (−1.27 − 0.737i)17-s + (−0.469 − 0.813i)19-s + (0.151 − 0.415i)20-s + (−0.242 − 0.203i)22-s + (0.262 + 0.720i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.16534 - 0.425874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16534 - 0.425874i\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.31 + 0.408i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.71 + 4.42i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.57 - 1.66i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (1.90 + 2.27i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (3.38 - 19.1i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (21.7 + 12.5i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (8.92 + 15.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.02 - 16.5i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-3.42 + 0.603i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-47.5 + 17.3i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (11.1 - 19.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.61 - 0.461i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-19.4 + 16.3i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-7.07 + 19.4i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 12.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-13.4 + 15.9i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-86.5 - 31.5i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-9.10 + 51.6i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (77.7 + 44.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (6.60 + 11.4i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.53 + 8.71i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-33.4 + 5.89i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 7.31i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (82.6 - 69.3i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−13.60370977125912557187546085549, −13.38208251898864666569773774018, −11.99400443859273286303855306779, −11.29126390656497618397142647508, −9.376309646296003966859831327856, −8.637868870996177540454398745422, −6.59489846007757768348127386537, −5.16546588353924272598539191500, −4.44570844280810527331457566205, −2.25127847448483627478023089194,
2.70014243564161885707780283794, 4.40592390282204541414519551166, 5.73311644523627538375517490520, 6.73547560415167030141703835018, 8.339139545848849558751989709425, 10.07207887014152221760834935571, 10.87689708390684144501859665384, 12.43707817315579154117805612553, 13.24997204528591766822478895957, 14.27854083868271680693148880413
