| L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.127 + 1.72i)3-s + (−0.733 − 0.680i)4-s + (−2.77 + 1.33i)5-s + (−1.56 − 0.749i)6-s + (−0.904 − 2.48i)7-s + (0.900 − 0.433i)8-s + (−2.96 − 0.440i)9-s + (−0.230 − 3.07i)10-s + (0.415 + 0.520i)11-s + (1.26 − 1.17i)12-s + (1.15 − 2.95i)13-s + (2.64 + 0.0667i)14-s + (−1.95 − 4.96i)15-s + (0.0747 + 0.997i)16-s + (1.43 − 1.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 + 0.658i)2-s + (−0.0736 + 0.997i)3-s + (−0.366 − 0.340i)4-s + (−1.24 + 0.598i)5-s + (−0.637 − 0.306i)6-s + (−0.341 − 0.939i)7-s + (0.318 − 0.153i)8-s + (−0.989 − 0.146i)9-s + (−0.0728 − 0.972i)10-s + (0.125 + 0.156i)11-s + (0.366 − 0.340i)12-s + (0.321 − 0.818i)13-s + (0.706 + 0.0178i)14-s + (−0.505 − 1.28i)15-s + (0.0186 + 0.249i)16-s + (0.348 − 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.641933 + 0.0417020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641933 + 0.0417020i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.365 - 0.930i)T \) |
| 3 | \( 1 + (0.127 - 1.72i)T \) |
| 7 | \( 1 + (0.904 + 2.48i)T \) |
| good | 5 | \( 1 + (2.77 - 1.33i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-0.415 - 0.520i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 2.95i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.43 + 1.33i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.384 + 0.666i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.675 - 2.95i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-4.22 - 1.30i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-0.783 + 1.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.99 + 1.23i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (2.55 + 1.74i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-5.72 + 3.90i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-0.673 + 1.71i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-8.47 + 2.61i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (6.42 - 4.37i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-0.792 + 0.735i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 + 12.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.70 + 11.8i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (12.3 + 1.85i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-4.52 - 7.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.30 + 3.33i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (1.67 + 4.26i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (2.63 - 4.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.26601042355174004973562173491, −9.334404049349351430776372889213, −8.352140997753666874589280638130, −7.60192694484225327854336133854, −6.91742179499655891234500420130, −5.81356806226700411666493754300, −4.74001211108218855383610705072, −3.81727142827061333094156728330, −3.19842555035532926577170736250, −0.42300680841149201000447526204,
1.07765548465097907976981467151, 2.42764090269589518904605363602, 3.54314709659478096461028814492, 4.62993036896719344773607018818, 5.83647172780112042010557446979, 6.80326519117144174597651668044, 7.82551779449223983975873883553, 8.559908827546924062043549482590, 8.921552776496004876364191367550, 10.16994961698897123300071085191
