| L(s) = 1 | + (−0.870 + 0.492i)2-s + (2.46 + 2.30i)3-s + (0.515 − 0.856i)4-s + (−2.32 − 0.0854i)5-s + (−3.27 − 0.798i)6-s + (−0.439 − 2.24i)7-s + (−0.0275 + 0.999i)8-s + (0.539 + 8.37i)9-s + (2.06 − 1.06i)10-s + (3.75 + 5.10i)11-s + (3.24 − 0.918i)12-s + (0.512 + 0.219i)13-s + (1.48 + 1.74i)14-s + (−5.52 − 5.57i)15-s + (−0.467 − 0.883i)16-s + (−4.50 + 0.0827i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.347i)2-s + (1.42 + 1.33i)3-s + (0.257 − 0.428i)4-s + (−1.03 − 0.0382i)5-s + (−1.33 − 0.326i)6-s + (−0.166 − 0.850i)7-s + (−0.00974 + 0.353i)8-s + (0.179 + 2.79i)9-s + (0.653 − 0.338i)10-s + (1.13 + 1.53i)11-s + (0.937 − 0.265i)12-s + (0.142 + 0.0608i)13-s + (0.397 + 0.465i)14-s + (−1.42 − 1.43i)15-s + (−0.116 − 0.220i)16-s + (−1.09 + 0.0200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436211 + 1.32741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436211 + 1.32741i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.870 - 0.492i)T \) |
| 19 | \( 1 + (-4.09 - 1.50i)T \) |
| good | 3 | \( 1 + (-2.46 - 2.30i)T + (0.192 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.32 + 0.0854i)T + (4.98 + 0.367i)T^{2} \) |
| 7 | \( 1 + (0.439 + 2.24i)T + (-6.48 + 2.63i)T^{2} \) |
| 11 | \( 1 + (-3.75 - 5.10i)T + (-3.28 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.512 - 0.219i)T + (8.97 + 9.40i)T^{2} \) |
| 17 | \( 1 + (4.50 - 0.0827i)T + (16.9 - 0.624i)T^{2} \) |
| 23 | \( 1 + (3.21 + 0.972i)T + (19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (-3.30 - 2.66i)T + (6.08 + 28.3i)T^{2} \) |
| 31 | \( 1 + (0.957 + 2.55i)T + (-23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (4.68 - 10.6i)T + (-25.0 - 27.2i)T^{2} \) |
| 41 | \( 1 + (5.56 + 0.512i)T + (40.3 + 7.49i)T^{2} \) |
| 43 | \( 1 + (0.511 + 11.1i)T + (-42.8 + 3.94i)T^{2} \) |
| 47 | \( 1 + (-0.720 + 0.754i)T + (-2.15 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-1.47 - 5.04i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (-0.846 - 1.83i)T + (-38.3 + 44.8i)T^{2} \) |
| 61 | \( 1 + (-9.34 + 1.91i)T + (56.0 - 23.9i)T^{2} \) |
| 67 | \( 1 + (-6.33 + 0.937i)T + (64.1 - 19.4i)T^{2} \) |
| 71 | \( 1 + (-2.23 - 0.458i)T + (65.2 + 27.9i)T^{2} \) |
| 73 | \( 1 + (-0.604 + 1.09i)T + (-38.7 - 61.8i)T^{2} \) |
| 79 | \( 1 + (0.524 - 0.336i)T + (33.0 - 71.7i)T^{2} \) |
| 83 | \( 1 + (-0.531 + 3.83i)T + (-79.8 - 22.5i)T^{2} \) |
| 89 | \( 1 + (-5.22 - 9.44i)T + (-47.3 + 75.3i)T^{2} \) |
| 97 | \( 1 + (1.22 + 0.182i)T + (92.8 + 28.1i)T^{2} \) |
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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
−10.16708068737575167012288930277, −9.956707893089622000993721012918, −8.957201620016834571226521012999, −8.374923291601685049636549472105, −7.45352835185515115638299547589, −6.89055058340990021251335959971, −4.88872841111000773000316818660, −4.14156773735076318913863896321, −3.53852677328416100139940741037, −1.96425316196458162742809629114,
0.76327219865020930964218531639, 2.14459538056942022900539385672, 3.22996497053693173517623308701, 3.81349487298106678026727756182, 6.07768220997548838942857695580, 6.85804588990154080021602366584, 7.72437004040637100695212713810, 8.545494870471720900891529678367, 8.794121707870999299381704011819, 9.607310652455295728641460123625
