L(s) = 1 | + 2.61·2-s + 4.85·4-s + (−1.61 − 2.80i)5-s + (−0.118 + 0.204i)7-s + 7.47·8-s + (−4.23 − 7.33i)10-s + 1.38·11-s + (−1.80 + 3.13i)13-s + (−0.309 + 0.535i)14-s + 9.85·16-s + (−2.54 + 4.40i)17-s + (−1.61 − 2.80i)19-s + (−7.85 − 13.6i)20-s + 3.61·22-s + (3.30 + 5.73i)23-s + ⋯ |
L(s) = 1 | + 1.85·2-s + 2.42·4-s + (−0.723 − 1.25i)5-s + (−0.0446 + 0.0772i)7-s + 2.64·8-s + (−1.33 − 2.32i)10-s + 0.416·11-s + (−0.501 + 0.869i)13-s + (−0.0825 + 0.143i)14-s + 2.46·16-s + (−0.617 + 1.06i)17-s + (−0.371 − 0.642i)19-s + (−1.75 − 3.04i)20-s + 0.771·22-s + (0.689 + 1.19i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.38805 - 0.752957i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.38805 - 0.752957i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (6.5 + 0.866i)T \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 + (1.61 + 2.80i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + (1.80 - 3.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.54 - 4.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.30 - 5.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.927 + 1.60i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.527T + 41T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 + (1.80 + 3.13i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + (-1.92 + 3.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.42 + 2.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.89 + 11.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.42 + 4.20i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.80 + 3.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.51 + 11.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.42 - 4.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−11.68979292233585407633965676577, −10.95001334384411482794288039466, −9.388219390513786409918144559829, −8.326413393171238875174215777658, −7.14967501520449008266804408669, −6.24092857516818876140666146577, −5.02367580718238670301237165275, −4.43861141019056810970362673115, −3.53333195714306311824259795575, −1.85514563682022815991985534519,
2.56327692664283757678581075940, 3.31863811460991859938183471188, 4.36408407197613184849446283523, 5.39760987428249841793093406373, 6.77415307266928779591835410604, 6.93512064305874472716357360859, 8.216962017218923593468545866903, 10.08947589812656564809199197520, 10.92287633043037631902236927124, 11.55292437840667668480482209505