| 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
−10.87032887718683077327542242991, −9.978861987749036156969071874164, −8.938143038616354677091165272581, −8.298397239065127328204319824052, −7.10386310185434440461614902124, −6.24213158994316812049482426419, −5.52000803560160191565555986907, −3.98367104004085552892740308536, −3.36557782873935563379190374720, −1.53699247410044867099137980128,
0.01581851820850923725647747300, 2.49118565687535799331163023530, 3.59631653429059182146132554774, 4.27346643326743745458572463451, 5.75694456202802376182871788079, 6.76752645750224311626472775743, 7.60438627126361097064069893666, 8.275174000805128993972271535712, 9.089747332114070997038001626023, 9.948696629873321994180370095553
