| L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (1.01 − 1.76i)5-s + (0.499 + 0.866i)6-s + (−0.0180 + 0.171i)7-s + (0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (1.36 + 1.51i)10-s + (4.46 − 1.98i)11-s + (−0.978 + 0.207i)12-s + (−3.16 − 0.672i)13-s + (−0.157 − 0.0703i)14-s + (−0.629 − 1.93i)15-s + (0.309 + 0.951i)16-s + (6.29 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.386 − 0.429i)3-s + (−0.404 − 0.293i)4-s + (0.455 − 0.788i)5-s + (0.204 + 0.353i)6-s + (−0.00683 + 0.0649i)7-s + (0.286 − 0.207i)8-s + (−0.0348 − 0.331i)9-s + (0.430 + 0.478i)10-s + (1.34 − 0.599i)11-s + (−0.282 + 0.0600i)12-s + (−0.877 − 0.186i)13-s + (−0.0422 − 0.0187i)14-s + (−0.162 − 0.500i)15-s + (0.0772 + 0.237i)16-s + (1.52 + 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23465 - 0.0752622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23465 - 0.0752622i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 31 | \( 1 + (2.76 - 4.83i)T \) |
| good | 5 | \( 1 + (-1.01 + 1.76i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.0180 - 0.171i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-4.46 + 1.98i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.16 + 0.672i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-6.29 - 2.80i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (3.16 - 0.672i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (4.85 - 3.52i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.760 + 2.34i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.547 + 0.947i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.25 + 8.06i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (6.58 - 1.39i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.46 - 7.57i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.413 + 3.93i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (4.44 - 4.93i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 + (1.46 - 2.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.336 - 3.20i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (1.19 - 0.532i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.02 - 2.68i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.53 - 5.03i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-8.72 - 6.33i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.39 - 6.82i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.59345117736065060164160290619, −11.95751548110343372796413890661, −10.26185817947035427643350077825, −9.305243252346202617746059944450, −8.534550416048877531554425485465, −7.54490630036161903001886221648, −6.29440201282296186240870378272, −5.35412437144528816060227935443, −3.74566622152140132278035719941, −1.48108711313191562157327269365,
2.12214669556417009960019913563, 3.46474522662023914138701055587, 4.73904347692009456552378970665, 6.43485018537703432022466070426, 7.58663238088928289407119729376, 8.928552962846522968545725109363, 9.905322361283083203077330557264, 10.30797997432646187193786511341, 11.68798407319558967083355519191, 12.32812956629603351072652211819
