| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.736 + 0.818i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (3.09 − 0.657i)11-s + (−0.104 − 0.994i)12-s + (0.424 − 4.03i)13-s + (1.07 + 0.228i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−4.08 − 0.867i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (0.278 + 0.309i)7-s + (−0.109 − 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (0.933 − 0.198i)11-s + (−0.0301 − 0.287i)12-s + (0.117 − 1.11i)13-s + (0.287 + 0.0611i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.989 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00406 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.83558 - 1.84307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83558 - 1.84307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.49 + 0.888i)T \) |
| good | 7 | \( 1 + (-0.736 - 0.818i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.09 + 0.657i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.424 + 4.03i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (4.08 + 0.867i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.190 + 1.81i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.98 - 6.09i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-8.38 + 6.09i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1.56 - 2.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.76 - 2.12i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.0293 + 0.279i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-0.876 - 0.636i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.76 + 5.28i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (5.24 - 2.33i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 + (-0.937 - 1.62i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.86 - 9.84i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (11.8 - 2.51i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-11.6 - 2.47i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 4.97i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (4.83 - 14.8i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.49 - 10.7i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.795944214467386634896601307919, −9.010913765539887561494312751718, −8.308354566915364312069348813596, −7.31775161431055607952040322227, −6.33410196084399958294813935997, −5.35967642117599741901092359383, −4.38366036426877871502886378083, −3.43668865022805137554130885342, −2.38426378214641498003581281461, −1.04808766042099698830353131760,
1.83224188239811593877112208207, 3.11852885545446698296185565103, 4.22709139353093634089609622735, 4.61228247661002799607336595329, 6.17951573318734677140957340250, 6.84565031639372595597672867871, 7.55500409686391502471902481621, 8.796297853786807269833538876987, 9.035845577348477153119296762445, 10.45968424829536706907519433651
