L(s) = 1 | + (0.866 − 0.5i)2-s + (0.0292 − 1.73i)3-s + (0.499 − 0.866i)4-s + 5-s + (−0.840 − 1.51i)6-s + (2.63 − 0.194i)7-s − 0.999i·8-s + (−2.99 − 0.101i)9-s + (0.866 − 0.5i)10-s − 2.09i·11-s + (−1.48 − 0.891i)12-s + (−0.413 + 0.238i)13-s + (2.18 − 1.48i)14-s + (0.0292 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−1.44 − 2.49i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.0168 − 0.999i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (−0.343 − 0.618i)6-s + (0.997 − 0.0735i)7-s − 0.353i·8-s + (−0.999 − 0.0337i)9-s + (0.273 − 0.158i)10-s − 0.630i·11-s + (−0.428 − 0.257i)12-s + (−0.114 + 0.0662i)13-s + (0.584 − 0.397i)14-s + (0.00755 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.349 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44254 - 1.87928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44254 - 1.87928i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.0292 + 1.73i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.63 + 0.194i)T \) |
good | 11 | \( 1 + 2.09iT - 11T^{2} \) |
| 13 | \( 1 + (0.413 - 0.238i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.44 + 2.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.03 - 2.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.02iT - 23T^{2} \) |
| 29 | \( 1 + (-2.74 - 1.58i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.59 + 2.65i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.93 - 3.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.03 + 6.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.96 - 5.14i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.0525 - 0.0909i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.27 - 2.46i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.40 + 2.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 2.14i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.73 - 4.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 + (-5.64 + 3.26i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.81 - 13.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.58 + 9.68i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.59 - 11.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.98 - 4.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.65585089843323142032108193126, −9.480056527392162554560288967988, −8.483896203939299605875357489395, −7.59706416638103797723520768035, −6.70944902769173677529855084396, −5.65302765647583570950853242548, −5.01142229494001997659341395062, −3.47249443335335457880344311594, −2.27480498177709257868049898829, −1.19570272765458749265553912613,
2.09055837399166878201390789908, 3.41511366746878111593334161046, 4.62920623939126711867058344931, 5.09114570312642102691588503160, 6.10431982784578331494365919049, 7.26084386957075662369570688970, 8.296707505320967532835130065171, 9.074195781472060017067994189787, 10.06384554050094274794753615099, 10.85547320523706285255527346033