| 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 | \( 1 + (-3.24 + 2.81i)T + (13.8 - 96.0i)T^{2} \) |
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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
−14.23853046128078352346731319267, −13.75374961393653118608397078528, −12.40533598833556161628366404832, −11.06962585393885949498262365507, −9.256213579052269907285696975184, −8.340517423796097276391225870094, −7.72334660618335970756772908688, −6.40456715677746036514743919782, −4.72926229481682348132017256107, −2.88266816260284670210028246568,
2.27282752084962362412821926846, 4.00294090932685059375795407283, 4.82415942749462416984295940115, 7.71249699776801153223336948074, 8.460130170548704165902577841040, 9.759897317207274418110936947410, 10.40118004125477619741502054884, 11.91368242812691401116835330986, 12.71991526058025249032069050317, 13.95829897538232068935995516089
