L(s) = 1 | + (1.27 + 2.20i)2-s + 3-s + (−2.25 + 3.90i)4-s + (−1.40 + 2.44i)5-s + (1.27 + 2.20i)6-s + (−0.0337 − 2.64i)7-s − 6.39·8-s + 9-s − 7.18·10-s + 0.531·11-s + (−2.25 + 3.90i)12-s + (2.71 − 2.36i)13-s + (5.80 − 3.44i)14-s + (−1.40 + 2.44i)15-s + (−3.64 − 6.31i)16-s + (−0.202 + 0.351i)17-s + ⋯ |
L(s) = 1 | + (0.901 + 1.56i)2-s + 0.577·3-s + (−1.12 + 1.95i)4-s + (−0.630 + 1.09i)5-s + (0.520 + 0.901i)6-s + (−0.0127 − 0.999i)7-s − 2.26·8-s + 0.333·9-s − 2.27·10-s + 0.160·11-s + (−0.650 + 1.12i)12-s + (0.753 − 0.657i)13-s + (1.55 − 0.921i)14-s + (−0.363 + 0.630i)15-s + (−0.911 − 1.57i)16-s + (−0.0492 + 0.0852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.525310 + 1.96602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.525310 + 1.96602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (0.0337 + 2.64i)T \) |
| 13 | \( 1 + (-2.71 + 2.36i)T \) |
good | 2 | \( 1 + (-1.27 - 2.20i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.40 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 0.531T + 11T^{2} \) |
| 17 | \( 1 + (0.202 - 0.351i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 + (-1.27 - 2.20i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.44 - 7.69i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.44 + 7.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.47 - 7.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.44 + 9.43i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.52 + 7.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.45 + 7.71i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.51 + 6.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.0364 - 0.0631i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 - 7.96T + 67T^{2} \) |
| 71 | \( 1 + (0.535 + 0.928i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.729 - 1.26i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.53 + 6.12i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.449T + 83T^{2} \) |
| 89 | \( 1 + (-0.461 - 0.799i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0841 + 0.145i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.83489891701929856556510488929, −11.44345616775643393257108567426, −10.45821047592643814525182493093, −9.026080509650121521005134345720, −7.80359000314626740180174058011, −7.38344398122879257232769469533, −6.58369901322561590776435770011, −5.32290298845259193447819427635, −3.81858685557187368807502000890, −3.41290111510347647346235607714,
1.38996193173241685078806458794, 2.79872242060875363129434845830, 4.00297806871824021210119016665, 4.82822103773292254524049816349, 5.98609284880407188535587846183, 7.980210661136511261216061963307, 9.142958804149632001933685549548, 9.440835174295761408928090757368, 11.02292837785769936934862189793, 11.65648825084692444029518742447