| L(s) = 1 | + (0.369 + 2.09i)2-s + (−2.36 + 0.860i)4-s + (1.58 + 1.33i)5-s + (−4.55 − 1.65i)7-s + (−0.547 − 0.949i)8-s + (−2.20 + 3.81i)10-s + (−3.17 + 2.66i)11-s + (−0.211 + 1.19i)13-s + (1.78 − 10.1i)14-s + (−2.07 + 1.73i)16-s + (1.18 − 2.04i)17-s + (0.919 + 1.59i)19-s + (−4.89 − 1.78i)20-s + (−6.75 − 5.66i)22-s + (−4.04 + 1.47i)23-s + ⋯ |
| L(s) = 1 | + (0.260 + 1.47i)2-s + (−1.18 + 0.430i)4-s + (0.709 + 0.595i)5-s + (−1.72 − 0.626i)7-s + (−0.193 − 0.335i)8-s + (−0.695 + 1.20i)10-s + (−0.958 + 0.804i)11-s + (−0.0585 + 0.332i)13-s + (0.478 − 2.71i)14-s + (−0.517 + 0.434i)16-s + (0.286 − 0.496i)17-s + (0.210 + 0.365i)19-s + (−1.09 − 0.398i)20-s + (−1.44 − 1.20i)22-s + (−0.843 + 0.306i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.397077 - 0.790646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.397077 - 0.790646i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.369 - 2.09i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.58 - 1.33i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (4.55 + 1.65i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (3.17 - 2.66i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.211 - 1.19i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.18 + 2.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.04 - 1.47i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.517 + 2.93i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.38 - 0.503i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.48 - 7.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.392 - 2.22i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.20 - 3.52i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.74 - 2.45i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + (0.200 + 0.168i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.18 - 1.52i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.717 + 4.06i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 2.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.38 + 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.790 - 4.48i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.46 - 8.32i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-8.48 - 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.91 - 3.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.47935777625814602194503964788, −9.985900012259559618651645061976, −9.295467969104318901433857892718, −7.951279555523076673881129822919, −7.22778199303350292765241661017, −6.54073224219121806924929399112, −5.96793381710603951921464794304, −4.93827729700542470259261268725, −3.70905640378770869341272450910, −2.44720108802644517258148956026,
0.38624318065652344274110911486, 2.09329264537783520884820855872, 3.01197878052756478087064340736, 3.79927227442614578427363887847, 5.36068159972908985225937140399, 5.84542930405431537416641735925, 7.13466159511818712930508239356, 8.678185611874400495755952632206, 9.211029676444127814661691149473, 10.18237480623253846259298042277
