| L(s) = 1 | + (−0.824 − 0.154i)2-s + (−0.224 − 0.451i)3-s + (−1.20 − 0.468i)4-s + (−2.39 + 1.48i)5-s + (0.115 + 0.406i)6-s + (−0.383 + 0.507i)7-s + (2.35 + 1.45i)8-s + (1.65 − 2.19i)9-s + (2.20 − 0.852i)10-s + (2.02 + 0.785i)11-s + (0.0603 + 0.650i)12-s + (3.58 + 2.21i)13-s + (0.394 − 0.359i)14-s + (1.20 + 0.747i)15-s + (0.202 + 0.184i)16-s + (1.56 − 3.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.583 − 0.108i)2-s + (−0.129 − 0.260i)3-s + (−0.604 − 0.234i)4-s + (−1.07 + 0.662i)5-s + (0.0472 + 0.166i)6-s + (−0.144 + 0.191i)7-s + (0.831 + 0.514i)8-s + (0.551 − 0.730i)9-s + (0.696 − 0.269i)10-s + (0.611 + 0.236i)11-s + (0.0174 + 0.187i)12-s + (0.994 + 0.615i)13-s + (0.105 − 0.0960i)14-s + (0.311 + 0.192i)15-s + (0.0505 + 0.0460i)16-s + (0.378 − 0.925i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.710598 + 0.0175549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.710598 + 0.0175549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-1.56 + 3.81i)T \) |
| good | 2 | \( 1 + (0.824 + 0.154i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (0.224 + 0.451i)T + (-1.80 + 2.39i)T^{2} \) |
| 5 | \( 1 + (2.39 - 1.48i)T + (2.22 - 4.47i)T^{2} \) |
| 7 | \( 1 + (0.383 - 0.507i)T + (-1.91 - 6.73i)T^{2} \) |
| 11 | \( 1 + (-2.02 - 0.785i)T + (8.12 + 7.41i)T^{2} \) |
| 13 | \( 1 + (-3.58 - 2.21i)T + (5.79 + 11.6i)T^{2} \) |
| 19 | \( 1 + (-5.46 + 1.02i)T + (17.7 - 6.86i)T^{2} \) |
| 23 | \( 1 + (3.57 - 4.73i)T + (-6.29 - 22.1i)T^{2} \) |
| 29 | \( 1 + (0.635 + 0.246i)T + (21.4 + 19.5i)T^{2} \) |
| 31 | \( 1 + (-3.73 - 2.31i)T + (13.8 + 27.7i)T^{2} \) |
| 37 | \( 1 + (-0.0551 + 0.595i)T + (-36.3 - 6.79i)T^{2} \) |
| 41 | \( 1 + (-4.85 - 9.74i)T + (-24.7 + 32.7i)T^{2} \) |
| 43 | \( 1 + (-8.75 + 7.98i)T + (3.96 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-4.79 - 6.35i)T + (-12.8 + 45.2i)T^{2} \) |
| 53 | \( 1 + (6.20 - 8.22i)T + (-14.5 - 50.9i)T^{2} \) |
| 59 | \( 1 + (-1.91 + 6.71i)T + (-50.1 - 31.0i)T^{2} \) |
| 61 | \( 1 + (3.42 + 12.0i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (-9.00 + 1.68i)T + (62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (-2.35 + 3.12i)T + (-19.4 - 68.2i)T^{2} \) |
| 73 | \( 1 + (8.11 + 7.40i)T + (6.73 + 72.6i)T^{2} \) |
| 79 | \( 1 + (5.58 - 1.04i)T + (73.6 - 28.5i)T^{2} \) |
| 83 | \( 1 + (6.69 - 13.4i)T + (-50.0 - 66.2i)T^{2} \) |
| 89 | \( 1 + (-5.03 + 3.11i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (-1.64 + 2.17i)T + (-26.5 - 93.2i)T^{2} \) |
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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
−11.64304288162513960369339525023, −11.01070974411222490925129281880, −9.617847887766794066354697135593, −9.250529427003897041343727758801, −7.86676507610523160147269980011, −7.19795655012681133948050092707, −6.00142990664535717045725460064, −4.41095980013281081192826383056, −3.44086301971166781982840380776, −1.13330983443056249767971047929,
0.946334164621396566342524178795, 3.77489203917342304982206056072, 4.31606667406555981359682187683, 5.69411454502365430755677511442, 7.35760958975519766625904527284, 8.126843591210211798093036554394, 8.721061974353481039723308598333, 9.918137578255705693581475127309, 10.65309814974535559867927666540, 11.80194144582716366218271587284
