| L(s) = 1 | + (0.0693 − 0.231i)2-s + (1.62 + 1.06i)4-s + (1.68 − 3.90i)5-s + (0.0669 + 1.15i)7-s + (0.729 − 0.612i)8-s + (−0.786 − 0.660i)10-s + (−1.17 + 1.58i)11-s + (2.74 − 2.91i)13-s + (0.271 + 0.0642i)14-s + (1.44 + 3.35i)16-s + (0.731 − 4.14i)17-s + (0.525 + 2.97i)19-s + (6.89 − 4.53i)20-s + (0.285 + 0.382i)22-s + (−0.0709 + 1.21i)23-s + ⋯ |
| L(s) = 1 | + (0.0490 − 0.163i)2-s + (0.811 + 0.533i)4-s + (0.752 − 1.74i)5-s + (0.0253 + 0.434i)7-s + (0.258 − 0.216i)8-s + (−0.248 − 0.208i)10-s + (−0.355 + 0.477i)11-s + (0.762 − 0.807i)13-s + (0.0724 + 0.0171i)14-s + (0.361 + 0.838i)16-s + (0.177 − 1.00i)17-s + (0.120 + 0.683i)19-s + (1.54 − 1.01i)20-s + (0.0607 + 0.0816i)22-s + (−0.0147 + 0.253i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00648 - 0.838126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00648 - 0.838126i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.0693 + 0.231i)T + (-1.67 - 1.09i)T^{2} \) |
| 5 | \( 1 + (-1.68 + 3.90i)T + (-3.43 - 3.63i)T^{2} \) |
| 7 | \( 1 + (-0.0669 - 1.15i)T + (-6.95 + 0.812i)T^{2} \) |
| 11 | \( 1 + (1.17 - 1.58i)T + (-3.15 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-2.74 + 2.91i)T + (-0.755 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.731 + 4.14i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.525 - 2.97i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.0709 - 1.21i)T + (-22.8 - 2.67i)T^{2} \) |
| 29 | \( 1 + (-0.580 + 0.137i)T + (25.9 - 13.0i)T^{2} \) |
| 31 | \( 1 + (1.73 - 0.873i)T + (18.5 - 24.8i)T^{2} \) |
| 37 | \( 1 + (4.95 - 1.80i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-1.64 - 5.48i)T + (-34.2 + 22.5i)T^{2} \) |
| 43 | \( 1 + (7.24 + 0.847i)T + (41.8 + 9.91i)T^{2} \) |
| 47 | \( 1 + (-0.390 - 0.195i)T + (28.0 + 37.6i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 3.87i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.97 + 6.68i)T + (-16.9 + 56.5i)T^{2} \) |
| 61 | \( 1 + (-8.81 + 5.79i)T + (24.1 - 56.0i)T^{2} \) |
| 67 | \( 1 + (-7.12 - 1.68i)T + (59.8 + 30.0i)T^{2} \) |
| 71 | \( 1 + (-3.66 - 3.07i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.52 + 4.63i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (3.61 - 12.0i)T + (-66.0 - 43.4i)T^{2} \) |
| 83 | \( 1 + (3.32 - 11.0i)T + (-69.3 - 45.6i)T^{2} \) |
| 89 | \( 1 + (8.43 - 7.07i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.49 - 5.77i)T + (-66.5 + 70.5i)T^{2} \) |
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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
−10.14068527732037258560786425572, −9.481192283777870635713170789689, −8.414514215546362398685042185862, −8.004872184034251429774431890770, −6.71737176736224263065797209702, −5.60289279924462609632967586632, −5.05383101923947808999336997907, −3.69856617495797832716075897185, −2.35460957618670137347151141733, −1.25817828816705564729266406209,
1.72710953215629143148482420570, 2.73568374170636953278787683562, 3.79216371125701458260217756933, 5.51168692389319896961081119647, 6.23796899299077184223426181977, 6.84915575291140079799587482373, 7.52785882440832023729987405887, 8.849365598152702448214326964912, 10.04371165565364125710722803164, 10.57384135963911025765994579041
