L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 1.5i)3-s + (−0.500 − 0.866i)4-s − 1.73i·6-s + (2.59 + 1.5i)7-s + 3i·8-s + (−1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (0.866 − 1.5i)12-s + (1.73 − i)13-s + (−1.5 − 2.59i)14-s + (0.500 − 0.866i)16-s + 4i·17-s + (2.59 − 1.5i)18-s + 8·19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (−0.250 − 0.433i)4-s − 0.707i·6-s + (0.981 + 0.566i)7-s + 1.06i·8-s + (−0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (0.249 − 0.433i)12-s + (0.480 − 0.277i)13-s + (−0.400 − 0.694i)14-s + (0.125 − 0.216i)16-s + 0.970i·17-s + (0.612 − 0.353i)18-s + 1.83·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04822 + 0.217409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04822 + 0.217409i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.92 + 4i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.79 - 4.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (1.73 + i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.84122935949544078252744170935, −11.12804855227929657346593998576, −10.30334993026600208030861775769, −9.375817695707002706557743251260, −8.578638647989805930346787557363, −7.912352711906511156333769662445, −5.75790584858429441523721573098, −5.01176078281149394015673956521, −3.49208418543190910432014017075, −1.77190946552616613835539648406,
1.29092685980446188214799950289, 3.27877565443137169676674328242, 4.70770831648353199562520420336, 6.53159072992239292963102843242, 7.51386289938755394923643911101, 7.968780208217688930806018493554, 9.062248999002875606993402376261, 9.864549335628471389833744011795, 11.50227629745020673884248933243, 12.07802479609158136582253854710