L(s) = 1 | + (−2.05 + 1.18i)2-s + (1.81 − 3.14i)4-s − 3.43·5-s + 3.86i·8-s + (7.05 − 4.07i)10-s + 0.313i·11-s + (−5.09 + 2.94i)13-s + (−0.958 − 1.65i)16-s + (0.476 + 0.825i)17-s + (−1.09 − 0.630i)19-s + (−6.23 + 10.7i)20-s + (−0.372 − 0.645i)22-s + 6.82i·23-s + 6.80·25-s + (6.98 − 12.0i)26-s + ⋯ |
L(s) = 1 | + (−1.45 + 0.838i)2-s + (0.907 − 1.57i)4-s − 1.53·5-s + 1.36i·8-s + (2.23 − 1.28i)10-s + 0.0946i·11-s + (−1.41 + 0.816i)13-s + (−0.239 − 0.414i)16-s + (0.115 + 0.200i)17-s + (−0.250 − 0.144i)19-s + (−1.39 + 2.41i)20-s + (−0.0794 − 0.137i)22-s + 1.42i·23-s + 1.36·25-s + (1.36 − 2.37i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2271751677\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2271751677\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.05 - 1.18i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 11 | \( 1 - 0.313iT - 11T^{2} \) |
| 13 | \( 1 + (5.09 - 2.94i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.476 - 0.825i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 + 0.630i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.82iT - 23T^{2} \) |
| 29 | \( 1 + (3.43 + 1.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.53 + 2.61i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.68 - 4.65i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0699 + 0.121i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.00 - 1.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 5.98i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.824 - 1.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.57 - 1.48i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (0.354 - 0.204i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.23 + 9.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.00 + 6.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.05 - 1.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 6.06i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.387234148124552249498137369342, −8.732293228540209261798380182362, −7.81311332246943067189967131447, −7.41263693507410336150300423615, −6.86756957852575170619015750299, −5.67529042179466390370245379692, −4.55417483780794122468866045773, −3.54728929991755132143312966751, −1.90334383035815955151241451034, −0.26122224468840269767490905146,
0.66534864241143390718086341167, 2.32707983596550185953352785545, 3.21176836532894740255638492395, 4.21295148982001905137252268442, 5.35161732049463603810015243592, 7.05150021223714849513411877181, 7.45753326636897297667079454313, 8.228208126534677316317110598182, 8.787563637548211470159270321116, 9.694411441942519047786666562103