| L(s) = 1 | + (−1.18 − 0.766i)2-s + (−0.871 − 1.90i)3-s + (0.826 + 1.82i)4-s + (−1.73 − 2.00i)5-s + (−0.425 + 2.93i)6-s + (0.195 + 0.125i)7-s + (0.413 − 2.79i)8-s + (−0.918 + 1.05i)9-s + (0.528 + 3.70i)10-s + (−5.40 + 0.777i)11-s + (2.75 − 3.16i)12-s + (0.562 + 0.876i)13-s + (−0.135 − 0.298i)14-s + (−2.30 + 5.05i)15-s + (−2.63 + 3.00i)16-s + (1.07 + 3.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.840 − 0.541i)2-s + (−0.503 − 1.10i)3-s + (0.413 + 0.910i)4-s + (−0.775 − 0.894i)5-s + (−0.173 + 1.19i)6-s + (0.0737 + 0.0473i)7-s + (0.146 − 0.989i)8-s + (−0.306 + 0.353i)9-s + (0.167 + 1.17i)10-s + (−1.63 + 0.234i)11-s + (0.795 − 0.913i)12-s + (0.156 + 0.242i)13-s + (−0.0363 − 0.0797i)14-s + (−0.595 + 1.30i)15-s + (−0.658 + 0.752i)16-s + (0.261 + 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0836756 + 0.305514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0836756 + 0.305514i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.18 + 0.766i)T \) |
| 23 | \( 1 + (1.59 + 4.52i)T \) |
| good | 3 | \( 1 + (0.871 + 1.90i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (1.73 + 2.00i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.195 - 0.125i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (5.40 - 0.777i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.562 - 0.876i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 3.66i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.315 - 1.07i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.188 + 0.641i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (7.20 + 3.29i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.74 + 7.78i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.98 + 3.44i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.46 + 2.95i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 3.52iT - 47T^{2} \) |
| 53 | \( 1 + (-4.45 - 2.86i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.52 + 4.19i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.83 - 8.40i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (14.5 + 2.09i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-6.17 - 0.887i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (7.57 + 2.22i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.74 - 1.76i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.93 - 4.27i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.81 - 0.829i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 10.2i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.20612744709420326121569982999, −11.16044393844487976560454219483, −10.22407658951192235664289596680, −8.780669767817890573631953735324, −7.928946536174931621635415237920, −7.29702637301305378162762939057, −5.79779258476152877766052471175, −4.10256071160248771709507676664, −2.08871955683294972647375757928, −0.38076643255556823031584654683,
3.05109767962842439084560096089, 4.81731477867261267035319746983, 5.77801504766935913715907399152, 7.30990098879981265230984201470, 7.939130182957190754154700927986, 9.399072022954418084875307846074, 10.31480065341891079629198062776, 10.94288539167750859663476581060, 11.55232996000289384730681667319, 13.33788750876158254785989680379
