| L(s) = 1 | + (−0.742 + 0.291i)2-s + (0.610 − 1.62i)3-s + (−0.999 + 0.927i)4-s + (−3.49 + 2.38i)5-s + (0.0193 + 1.38i)6-s + (−2.63 + 0.242i)7-s + (1.16 − 2.41i)8-s + (−2.25 − 1.97i)9-s + (1.90 − 2.78i)10-s + (−0.481 + 3.19i)11-s + (0.893 + 2.18i)12-s + (1.47 + 1.17i)13-s + (1.88 − 0.947i)14-s + (1.73 + 7.11i)15-s + (0.0439 − 0.586i)16-s + (0.330 − 0.101i)17-s + ⋯ |
| L(s) = 1 | + (−0.524 + 0.206i)2-s + (0.352 − 0.935i)3-s + (−0.499 + 0.463i)4-s + (−1.56 + 1.06i)5-s + (0.00792 + 0.563i)6-s + (−0.995 + 0.0917i)7-s + (0.411 − 0.854i)8-s + (−0.751 − 0.659i)9-s + (0.600 − 0.881i)10-s + (−0.145 + 0.963i)11-s + (0.258 + 0.631i)12-s + (0.410 + 0.327i)13-s + (0.503 − 0.253i)14-s + (0.446 + 1.83i)15-s + (0.0109 − 0.146i)16-s + (0.0801 − 0.0247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0449592 + 0.194626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0449592 + 0.194626i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.610 + 1.62i)T \) |
| 7 | \( 1 + (2.63 - 0.242i)T \) |
| good | 2 | \( 1 + (0.742 - 0.291i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (3.49 - 2.38i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (0.481 - 3.19i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.47 - 1.17i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.330 + 0.101i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (3.79 + 2.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.617 - 2.00i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-3.02 - 0.690i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (6.61 - 3.81i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.19 - 3.88i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (5.35 + 2.58i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.13 - 0.545i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.205 - 0.523i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.0176 - 0.0190i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (2.74 + 1.87i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (4.20 - 4.53i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.45 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.01 - 1.60i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (2.47 + 0.972i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (1.49 - 2.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.31 + 6.66i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.88 + 0.284i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 - 11.3iT - 97T^{2} \) |
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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
−13.28829813280391147256858506705, −12.49630518175733891912033778984, −11.70948481456167953381700807622, −10.38240835771462313795993476663, −9.061436382544657514420072403596, −8.109554766216268983886132889961, −7.16929275687118761181426548355, −6.69472297691494599937215667920, −4.06305230651432061289768217265, −3.01975026222598591623136170068,
0.22303834826907756818830921336, 3.51924018084637439064559553498, 4.45682285946373757238706772470, 5.74940406229918741549593082053, 7.979104054939657266410074057485, 8.596129196522447780424913525989, 9.370719231028812542303950909276, 10.53766308280817655618221403204, 11.30735855696947963412320883695, 12.62970436039646569324770663113
