L(s) = 1 | + (0.166 − 0.942i)3-s + (0.766 + 0.642i)5-s + (−2.28 − 3.95i)7-s + (1.95 + 0.712i)9-s + (−0.558 + 0.968i)11-s + (−0.897 − 5.08i)13-s + (0.733 − 0.615i)15-s + (6.80 − 2.47i)17-s + (−3.68 − 2.32i)19-s + (−4.10 + 1.49i)21-s + (−3.31 + 2.78i)23-s + (0.173 + 0.984i)25-s + (2.43 − 4.21i)27-s + (2.51 + 0.913i)29-s + (1.37 + 2.37i)31-s + ⋯ |
L(s) = 1 | + (0.0959 − 0.544i)3-s + (0.342 + 0.287i)5-s + (−0.862 − 1.49i)7-s + (0.652 + 0.237i)9-s + (−0.168 + 0.291i)11-s + (−0.248 − 1.41i)13-s + (0.189 − 0.158i)15-s + (1.64 − 0.600i)17-s + (−0.845 − 0.533i)19-s + (−0.895 + 0.325i)21-s + (−0.692 + 0.580i)23-s + (0.0347 + 0.196i)25-s + (0.468 − 0.810i)27-s + (0.466 + 0.169i)29-s + (0.246 + 0.426i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03427 - 0.867747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03427 - 0.867747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (3.68 + 2.32i)T \) |
good | 3 | \( 1 + (-0.166 + 0.942i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (2.28 + 3.95i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.558 - 0.968i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.897 + 5.08i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-6.80 + 2.47i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (3.31 - 2.78i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.51 - 0.913i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 2.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.491T + 37T^{2} \) |
| 41 | \( 1 + (-1.45 + 8.24i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.16 - 2.65i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.48 - 0.904i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (10.4 - 8.76i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.45 + 0.892i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.64 + 3.06i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.9 - 4.33i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.524 - 0.439i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.28 - 12.9i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.75 - 15.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.88 - 3.25i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.04 - 11.5i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.27 + 1.19i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.86633080798294394252442278684, −10.14850525000308014093716505442, −9.734976192664704238311977341589, −7.988294158918393362940429875132, −7.37568911914749074237357312641, −6.63567863802847576767648046217, −5.37735314378167535904351413239, −3.97332016460182174643352166363, −2.79711242316252060685475987649, −0.955094985757206389908485166138,
2.01827482783724259057924323816, 3.43948283746362730987512256110, 4.63057600504073444896097587221, 5.88372976576412431351238658605, 6.50029181965070014252621449764, 8.081056968919908048991417103137, 9.019799322077524048482038317831, 9.697507777119690418208865536160, 10.31333102717858020802935441548, 11.79646095315281768416555174791