| L(s) = 1 | + (0.952 + 1.04i)2-s + (−2.70 + 0.678i)3-s + (−0.186 + 1.99i)4-s + (2.24 + 1.06i)5-s + (−3.28 − 2.18i)6-s + (0.244 + 0.806i)7-s + (−2.25 + 1.70i)8-s + (4.22 − 2.25i)9-s + (1.02 + 3.36i)10-s + (−4.99 + 0.740i)11-s + (−0.845 − 5.51i)12-s + (−0.0958 + 0.267i)13-s + (−0.610 + 1.02i)14-s + (−6.80 − 1.35i)15-s + (−3.93 − 0.742i)16-s + (−4.22 + 0.841i)17-s + ⋯ |
| L(s) = 1 | + (0.673 + 0.739i)2-s + (−1.56 + 0.391i)3-s + (−0.0931 + 0.995i)4-s + (1.00 + 0.475i)5-s + (−1.34 − 0.891i)6-s + (0.0924 + 0.304i)7-s + (−0.798 + 0.601i)8-s + (1.40 − 0.752i)9-s + (0.325 + 1.06i)10-s + (−1.50 + 0.223i)11-s + (−0.244 − 1.59i)12-s + (−0.0265 + 0.0743i)13-s + (−0.163 + 0.273i)14-s + (−1.75 − 0.349i)15-s + (−0.982 − 0.185i)16-s + (−1.02 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.180280 + 1.02358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.180280 + 1.02358i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.952 - 1.04i)T \) |
| good | 3 | \( 1 + (2.70 - 0.678i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-2.24 - 1.06i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-0.244 - 0.806i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (4.99 - 0.740i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.0958 - 0.267i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (4.22 - 0.841i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-0.745 - 0.675i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-9.28 + 0.914i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-2.94 - 3.97i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-6.55 - 2.71i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.409 - 8.33i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-0.584 + 0.712i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.45 - 5.81i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-0.447 - 0.299i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (1.09 - 1.48i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (4.68 + 13.0i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-5.36 + 8.94i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-11.0 - 6.62i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-0.268 + 0.502i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (2.34 - 7.72i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (3.62 + 5.42i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.662 - 13.4i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (0.176 + 0.0173i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-5.59 + 13.4i)T + (-68.5 - 68.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62512544971473667836438153713, −11.45673625774996644243154364046, −10.71602343676963796313411490078, −9.829645071087617109833330107938, −8.422706606681876174261772277643, −6.89143891186804971357793139694, −6.31814113135518933080623230606, −5.21744857464570343106959041891, −4.82021533424764141901607445081, −2.75480011738703339341058217304,
0.797357508378774365360225905964, 2.45253472207072852913367048961, 4.66300539894337777437403127797, 5.34037749397421028815901528395, 6.07955494222984475588577637782, 7.19866875549404145519503392329, 9.041224390248094903459997458466, 10.23393009803906019200565252489, 10.79593633156182838521188953852, 11.56910293866303242872248241433
