| L(s) = 1 | + (0.0506 − 1.41i)2-s + (1.95 + 0.891i)3-s + (−1.99 − 0.143i)4-s + (−1.03 − 0.895i)5-s + (1.35 − 2.71i)6-s + (2.14 − 1.38i)7-s + (−0.303 + 2.81i)8-s + (1.04 + 1.21i)9-s + (−1.31 + 1.41i)10-s + (−0.745 + 5.18i)11-s + (−3.76 − 2.05i)12-s + (−3.68 − 2.37i)13-s + (−1.84 − 3.10i)14-s + (−1.21 − 2.66i)15-s + (3.95 + 0.571i)16-s + (−0.725 + 2.46i)17-s + ⋯ |
| L(s) = 1 | + (0.0358 − 0.999i)2-s + (1.12 + 0.514i)3-s + (−0.997 − 0.0715i)4-s + (−0.461 − 0.400i)5-s + (0.554 − 1.10i)6-s + (0.811 − 0.521i)7-s + (−0.107 + 0.994i)8-s + (0.349 + 0.403i)9-s + (−0.416 + 0.447i)10-s + (−0.224 + 1.56i)11-s + (−1.08 − 0.593i)12-s + (−1.02 − 0.657i)13-s + (−0.492 − 0.829i)14-s + (−0.314 − 0.688i)15-s + (0.989 + 0.142i)16-s + (−0.175 + 0.598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05165 - 0.544593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05165 - 0.544593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0506 + 1.41i)T \) |
| 23 | \( 1 + (1.53 - 4.54i)T \) |
| good | 3 | \( 1 + (-1.95 - 0.891i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (1.03 + 0.895i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.14 + 1.38i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.745 - 5.18i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.68 + 2.37i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.725 - 2.46i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (2.83 - 0.832i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.83 - 1.42i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-5.16 + 2.35i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (0.714 - 0.618i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.41 + 3.94i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.17 + 4.75i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 6.04iT - 47T^{2} \) |
| 53 | \( 1 + (6.78 + 10.5i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.90 - 4.51i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.40 - 0.643i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.03 - 14.1i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (6.01 - 0.864i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.9 + 4.10i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 0.739i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.21 - 7.17i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.34 + 2.89i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.24 - 2.81i)T + (13.8 + 96.0i)T^{2} \) |
| 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
−13.95829897538232068935995516089, −12.71991526058025249032069050317, −11.91368242812691401116835330986, −10.40118004125477619741502054884, −9.759897317207274418110936947410, −8.460130170548704165902577841040, −7.71249699776801153223336948074, −4.82415942749462416984295940115, −4.00294090932685059375795407283, −2.27282752084962362412821926846,
2.88266816260284670210028246568, 4.72926229481682348132017256107, 6.40456715677746036514743919782, 7.72334660618335970756772908688, 8.340517423796097276391225870094, 9.256213579052269907285696975184, 11.06962585393885949498262365507, 12.40533598833556161628366404832, 13.75374961393653118608397078528, 14.23853046128078352346731319267
