| L(s) = 1 | + (−1.71 + 0.201i)2-s + (0.971 − 0.230i)4-s + (1.23 + 0.618i)5-s + (−0.943 − 3.15i)7-s + (1.63 − 0.593i)8-s + (−2.24 − 0.815i)10-s + (−0.0119 + 0.204i)11-s + (3.13 − 4.21i)13-s + (2.25 + 5.22i)14-s + (−4.46 + 2.24i)16-s + (−3.88 + 3.26i)17-s + (−2.25 − 1.88i)19-s + (1.33 + 0.316i)20-s + (−0.0206 − 0.354i)22-s + (−1.07 + 3.60i)23-s + ⋯ |
| L(s) = 1 | + (−1.21 + 0.142i)2-s + (0.485 − 0.115i)4-s + (0.550 + 0.276i)5-s + (−0.356 − 1.19i)7-s + (0.576 − 0.209i)8-s + (−0.708 − 0.257i)10-s + (−0.00359 + 0.0618i)11-s + (0.870 − 1.16i)13-s + (0.602 + 1.39i)14-s + (−1.11 + 0.560i)16-s + (−0.942 + 0.791i)17-s + (−0.516 − 0.433i)19-s + (0.298 + 0.0708i)20-s + (−0.00440 − 0.0756i)22-s + (−0.224 + 0.751i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.343518 - 0.430394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.343518 - 0.430394i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.71 - 0.201i)T + (1.94 - 0.461i)T^{2} \) |
| 5 | \( 1 + (-1.23 - 0.618i)T + (2.98 + 4.01i)T^{2} \) |
| 7 | \( 1 + (0.943 + 3.15i)T + (-5.84 + 3.84i)T^{2} \) |
| 11 | \( 1 + (0.0119 - 0.204i)T + (-10.9 - 1.27i)T^{2} \) |
| 13 | \( 1 + (-3.13 + 4.21i)T + (-3.72 - 12.4i)T^{2} \) |
| 17 | \( 1 + (3.88 - 3.26i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.25 + 1.88i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (1.07 - 3.60i)T + (-19.2 - 12.6i)T^{2} \) |
| 29 | \( 1 + (-2.49 + 5.79i)T + (-19.9 - 21.0i)T^{2} \) |
| 31 | \( 1 + (3.48 + 3.69i)T + (-1.80 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 8.19i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (2.74 + 0.320i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-1.32 - 0.871i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (1.42 - 1.51i)T + (-2.73 - 46.9i)T^{2} \) |
| 53 | \( 1 + (4.51 + 7.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.863 + 14.8i)T + (-58.6 + 6.84i)T^{2} \) |
| 61 | \( 1 + (1.46 + 0.347i)T + (54.5 + 27.3i)T^{2} \) |
| 67 | \( 1 + (5.31 + 12.3i)T + (-45.9 + 48.7i)T^{2} \) |
| 71 | \( 1 + (4.39 + 1.59i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.3 + 5.56i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.47 + 0.172i)T + (76.8 - 18.2i)T^{2} \) |
| 83 | \( 1 + (14.1 - 1.65i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (-4.83 + 1.75i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.60 - 0.805i)T + (57.9 - 77.8i)T^{2} \) |
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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
−10.05686436515492532902226423730, −9.461247990268415710879183812915, −8.296004931140467903568822405805, −7.86457273262868267995958079023, −6.71773840408836941484009881105, −6.13639696684373988158370887419, −4.55405406985821047970919352778, −3.52999478186539203140637296324, −1.86659807386196248191023615696, −0.43394641191717559077586572062,
1.56058418888987041477682754908, 2.51787763146070842481532544143, 4.23922995399265887813084377849, 5.40230928097915967710425691882, 6.36663792014075701103928731639, 7.27406727829642409824469391640, 8.714063466450867709687450640341, 8.836383149992301481304322241055, 9.468454566643590344162465175589, 10.49283899057144498942992514189
