L(s) = 1 | − 3.60·5-s + (−1.60 − 2.10i)7-s + 6.02·11-s + (−2.55 + 4.42i)13-s + (−0.111 + 0.192i)17-s + (1.71 + 2.96i)19-s + 1.01·23-s + 7.98·25-s + (2.83 + 4.91i)29-s + (2.52 + 4.37i)31-s + (5.77 + 7.58i)35-s + (1.68 + 2.91i)37-s + (−0.0955 + 0.165i)41-s + (1.71 + 2.96i)43-s + (−1.03 + 1.79i)47-s + ⋯ |
L(s) = 1 | − 1.61·5-s + (−0.605 − 0.795i)7-s + 1.81·11-s + (−0.709 + 1.22i)13-s + (−0.0269 + 0.0466i)17-s + (0.392 + 0.680i)19-s + 0.212·23-s + 1.59·25-s + (0.526 + 0.912i)29-s + (0.453 + 0.784i)31-s + (0.976 + 1.28i)35-s + (0.277 + 0.479i)37-s + (−0.0149 + 0.0258i)41-s + (0.261 + 0.452i)43-s + (−0.150 + 0.261i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.824357 + 0.448796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.824357 + 0.448796i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.60 + 2.10i)T \) |
good | 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + (2.55 - 4.42i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.111 - 0.192i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.71 - 2.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + (-2.83 - 4.91i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.52 - 4.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 2.91i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0955 - 0.165i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.71 - 2.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.03 - 1.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.65 + 4.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.891 - 1.54i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.49 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 + (-6.30 + 10.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.30 - 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.59 - 6.23i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.44 - 7.70i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.72 + 4.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.54780057308003114976149906389, −9.522295667192686370604817540890, −8.824081932111293491532544402343, −7.77389582229998690578163446347, −6.95106350275539657406006036150, −6.50246130790417274744165201949, −4.66871354749928742270318509726, −4.01537380417197195346537830607, −3.28369633593271615613877450140, −1.19650168114697973915277308253,
0.57953602288632331290657034238, 2.74392936581130352495958283180, 3.69006829114479379005920110110, 4.56977426207243121540624880333, 5.82429502431118877233885140357, 6.83837556422427501824174691085, 7.61044654451732965620981096704, 8.518538845441989200436154434413, 9.243530821132546202638838758601, 10.13739816345950041087431250711