| L(s) = 1 | + (−0.217 + 0.230i)2-s + (0.110 + 1.89i)4-s + (−2.10 − 0.246i)5-s + (−2.27 + 1.14i)7-s + (−0.947 − 0.795i)8-s + (0.515 − 0.432i)10-s + (−1.21 + 2.80i)11-s + (5.37 − 1.27i)13-s + (0.231 − 0.773i)14-s + (−3.38 + 0.395i)16-s + (−0.745 − 4.22i)17-s + (−0.0105 + 0.0600i)19-s + (0.234 − 4.02i)20-s + (−0.384 − 0.891i)22-s + (−3.82 − 1.92i)23-s + ⋯ |
| L(s) = 1 | + (−0.153 + 0.163i)2-s + (0.0552 + 0.948i)4-s + (−0.942 − 0.110i)5-s + (−0.859 + 0.431i)7-s + (−0.335 − 0.281i)8-s + (0.163 − 0.136i)10-s + (−0.365 + 0.847i)11-s + (1.48 − 0.353i)13-s + (0.0618 − 0.206i)14-s + (−0.845 + 0.0988i)16-s + (−0.180 − 1.02i)17-s + (−0.00243 + 0.0137i)19-s + (0.0523 − 0.899i)20-s + (−0.0819 − 0.190i)22-s + (−0.797 − 0.400i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0135801 - 0.0259094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0135801 - 0.0259094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.217 - 0.230i)T + (-0.116 - 1.99i)T^{2} \) |
| 5 | \( 1 + (2.10 + 0.246i)T + (4.86 + 1.15i)T^{2} \) |
| 7 | \( 1 + (2.27 - 1.14i)T + (4.18 - 5.61i)T^{2} \) |
| 11 | \( 1 + (1.21 - 2.80i)T + (-7.54 - 8.00i)T^{2} \) |
| 13 | \( 1 + (-5.37 + 1.27i)T + (11.6 - 5.83i)T^{2} \) |
| 17 | \( 1 + (0.745 + 4.22i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.0105 - 0.0600i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (3.82 + 1.92i)T + (13.7 + 18.4i)T^{2} \) |
| 29 | \( 1 + (3.06 + 10.2i)T + (-24.2 + 15.9i)T^{2} \) |
| 31 | \( 1 + (3.85 - 2.53i)T + (12.2 - 28.4i)T^{2} \) |
| 37 | \( 1 + (3.30 + 1.20i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (1.72 + 1.83i)T + (-2.38 + 40.9i)T^{2} \) |
| 43 | \( 1 + (0.463 + 0.622i)T + (-12.3 + 41.1i)T^{2} \) |
| 47 | \( 1 + (-3.79 - 2.49i)T + (18.6 + 43.1i)T^{2} \) |
| 53 | \( 1 + (-0.986 - 1.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.85 - 6.62i)T + (-40.4 + 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.679 + 11.6i)T + (-60.5 - 7.08i)T^{2} \) |
| 67 | \( 1 + (-0.347 + 1.16i)T + (-55.9 - 36.8i)T^{2} \) |
| 71 | \( 1 + (10.9 - 9.17i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-7.44 - 6.24i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.10 - 1.17i)T + (-4.59 - 78.8i)T^{2} \) |
| 83 | \( 1 + (2.26 - 2.40i)T + (-4.82 - 82.8i)T^{2} \) |
| 89 | \( 1 + (10.7 + 9.02i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (6.97 - 0.815i)T + (94.3 - 22.3i)T^{2} \) |
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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
−9.948696629873321994180370095553, −9.089747332114070997038001626023, −8.275174000805128993972271535712, −7.60438627126361097064069893666, −6.76752645750224311626472775743, −5.75694456202802376182871788079, −4.27346643326743745458572463451, −3.59631653429059182146132554774, −2.49118565687535799331163023530, −0.01581851820850923725647747300,
1.53699247410044867099137980128, 3.36557782873935563379190374720, 3.98367104004085552892740308536, 5.52000803560160191565555986907, 6.24213158994316812049482426419, 7.10386310185434440461614902124, 8.298397239065127328204319824052, 8.938143038616354677091165272581, 9.978861987749036156969071874164, 10.87032887718683077327542242991
