| L(s) = 1 | + (0.879 + 2.03i)2-s + (0.875 − 1.49i)3-s + (−2.00 + 2.13i)4-s + (0.0265 − 0.455i)5-s + (3.81 + 0.469i)6-s + (−3.62 + 0.860i)7-s + (−1.93 − 0.705i)8-s + (−1.46 − 2.61i)9-s + (0.951 − 0.346i)10-s + (2.35 − 1.54i)11-s + (1.42 + 4.86i)12-s + (−5.39 + 0.630i)13-s + (−4.94 − 6.64i)14-s + (−0.657 − 0.438i)15-s + (0.0744 + 1.27i)16-s + (1.14 + 0.957i)17-s + ⋯ |
| L(s) = 1 | + (0.621 + 1.44i)2-s + (0.505 − 0.862i)3-s + (−1.00 + 1.06i)4-s + (0.0118 − 0.203i)5-s + (1.55 + 0.191i)6-s + (−1.37 + 0.325i)7-s + (−0.684 − 0.249i)8-s + (−0.489 − 0.872i)9-s + (0.300 − 0.109i)10-s + (0.709 − 0.466i)11-s + (0.411 + 1.40i)12-s + (−1.49 + 0.174i)13-s + (−1.32 − 1.77i)14-s + (−0.169 − 0.113i)15-s + (0.0186 + 0.319i)16-s + (0.276 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09369 + 0.670594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09369 + 0.670594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.875 + 1.49i)T \) |
| good | 2 | \( 1 + (-0.879 - 2.03i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.0265 + 0.455i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (3.62 - 0.860i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-2.35 + 1.54i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (5.39 - 0.630i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-1.14 - 0.957i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.28 + 2.75i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-4.73 - 1.12i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (2.29 - 3.07i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (1.07 + 3.57i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (1.40 - 7.99i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (2.06 - 4.79i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (7.86 + 3.95i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-2.95 + 9.88i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-0.641 + 1.11i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.41 - 5.53i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (3.89 + 4.13i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.04 - 2.74i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-8.71 + 3.17i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.970 - 0.353i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.44 + 3.35i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (0.740 + 1.71i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (0.0783 + 0.0285i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.0857 + 1.47i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−14.66905915644042759273679365689, −13.55321007034801754228321458565, −12.90926790599731432736811184445, −11.90601410618733595045567173967, −9.570037404847492264604444249830, −8.612051811861934324142892229420, −7.15939446511754753823197062097, −6.63461076947123589517078824419, −5.28164422731040433716053280965, −3.25907691176259324551950257583,
2.74834737170856407790977333577, 3.74144495366555083588338922633, 5.07980322741669666054000177697, 7.18653937497360698119000292733, 9.386487002134400031626745618054, 9.854925217177198188300947686665, 10.79201219039176838677323416034, 12.12619905003348740010660446179, 12.90503998374808933328969141512, 14.10047366886465170056952560794
