| L(s) = 1 | + (0.129 + 0.0359i)2-s + (−1.69 − 1.02i)4-s + (2.51 − 0.0976i)5-s + (4.08 + 0.638i)7-s + (−0.366 − 0.388i)8-s + (0.328 + 0.0778i)10-s + (−0.654 − 4.78i)11-s + (−0.189 + 0.276i)13-s + (0.504 + 0.229i)14-s + (1.81 + 3.44i)16-s + (−1.56 + 0.787i)17-s + (0.123 + 2.12i)19-s + (−4.36 − 2.41i)20-s + (0.0877 − 0.642i)22-s + (0.620 − 1.60i)23-s + ⋯ |
| L(s) = 1 | + (0.0913 + 0.0254i)2-s + (−0.848 − 0.512i)4-s + (1.12 − 0.0436i)5-s + (1.54 + 0.241i)7-s + (−0.129 − 0.137i)8-s + (0.103 + 0.0246i)10-s + (−0.197 − 1.44i)11-s + (−0.0526 + 0.0767i)13-s + (0.134 + 0.0613i)14-s + (0.453 + 0.860i)16-s + (−0.380 + 0.190i)17-s + (0.0283 + 0.487i)19-s + (−0.977 − 0.539i)20-s + (0.0186 − 0.136i)22-s + (0.129 − 0.335i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71859 - 0.612069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71859 - 0.612069i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.129 - 0.0359i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-2.51 + 0.0976i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-4.08 - 0.638i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.654 + 4.78i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (0.189 - 0.276i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.56 - 0.787i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.123 - 2.12i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-0.620 + 1.60i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-0.238 - 0.170i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-5.27 + 6.04i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-6.18 - 8.31i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-1.90 + 7.40i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (0.684 + 0.620i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (4.38 + 5.02i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-2.13 + 12.1i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.52 + 0.623i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (0.475 - 0.287i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 4.45i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-3.03 - 10.1i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (10.8 - 2.56i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (11.3 - 11.1i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.20 - 12.4i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (2.10 - 7.03i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-0.615 - 0.0238i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.22404243816809609994246016874, −9.486326382223987648672467377861, −8.404196976016670643724120888904, −8.226527984912673572930521328114, −6.44936482129508131903783035868, −5.62919584862181432307008890293, −5.09405448190912865738837788852, −4.00400555150647635157867434122, −2.32352429063195538382966788947, −1.11068006733504424329624811257,
1.53510033750068317411054984549, 2.68778828665969101427304520475, 4.50665261944917320018404642480, 4.73839279605340879924598354195, 5.83708367643159024824743556239, 7.22306781207494882375331179824, 7.87260314373811646589038507867, 8.871200675633341326579994674156, 9.584635116728913271160218149471, 10.34384516610544995604965773426
