L(s) = 1 | − 2.13·2-s + (1.01 − 1.76i)3-s + 2.56·4-s + (−1.08 + 1.88i)5-s + (−2.17 + 3.77i)6-s + (0.371 + 0.642i)7-s − 1.21·8-s + (−0.575 − 0.996i)9-s + (2.32 − 4.02i)10-s + 0.410·11-s + (2.61 − 4.52i)12-s + (−1.10 − 1.91i)13-s + (−0.792 − 1.37i)14-s + (2.21 + 3.83i)15-s − 2.54·16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − 1.51·2-s + (0.588 − 1.01i)3-s + 1.28·4-s + (−0.485 + 0.841i)5-s + (−0.888 + 1.53i)6-s + (0.140 + 0.242i)7-s − 0.428·8-s + (−0.191 − 0.332i)9-s + (0.734 − 1.27i)10-s + 0.123·11-s + (0.754 − 1.30i)12-s + (−0.307 − 0.531i)13-s + (−0.211 − 0.367i)14-s + (0.571 + 0.989i)15-s − 0.635·16-s + (0.121 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.767576 + 0.0994152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.767576 + 0.0994152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-6.55 - 0.310i)T \) |
good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 3 | \( 1 + (-1.01 + 1.76i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.08 - 1.88i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.371 - 0.642i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.410T + 11T^{2} \) |
| 13 | \( 1 + (1.10 + 1.91i)T + (-6.5 + 11.2i)T^{2} \) |
| 19 | \( 1 + (-3.07 + 5.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.88 - 6.72i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.16 - 7.22i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.08 - 5.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.44 - 5.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.165T + 41T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + (-3.48 + 6.03i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 + (5.48 + 9.50i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.52 + 6.11i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.52 - 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.63 - 4.57i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.28 - 3.94i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.21 - 5.57i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−10.36869159154888939856867265139, −9.365474050158481495644893185342, −8.594713140786708657218325168944, −7.84844867043820438788190191310, −7.18996661390584017145167702385, −6.79855792960117747119480084808, −5.19746924361335580643113499668, −3.34710714931922403065097341945, −2.35336035283431010728862017794, −1.16006826720581262957949762946,
0.74438667001373731688709171903, 2.35191595940169296753463153320, 4.03930527411279478363640721437, 4.49733991299611760978694708182, 6.06171685599220936692403450630, 7.47349943618599476295095703669, 8.002674504565114097252832933932, 8.963208469458432135983554090308, 9.234818009616860770997839768365, 10.21772784061531057945919101135