L(s) = 1 | + (−1.60 + 0.649i)3-s − 2.96·5-s + (2.38 − 1.14i)7-s + (2.15 − 2.08i)9-s − 4.72i·11-s + (3.54 − 2.04i)13-s + (4.76 − 1.92i)15-s + (0.835 + 1.44i)17-s + (−4.25 − 2.45i)19-s + (−3.08 + 3.38i)21-s − 4.91i·23-s + 3.82·25-s + (−2.11 + 4.74i)27-s + (0.238 + 0.137i)29-s + (−1.38 − 0.801i)31-s + ⋯ |
L(s) = 1 | + (−0.927 + 0.374i)3-s − 1.32·5-s + (0.901 − 0.433i)7-s + (0.719 − 0.694i)9-s − 1.42i·11-s + (0.981 − 0.566i)13-s + (1.23 − 0.497i)15-s + (0.202 + 0.350i)17-s + (−0.975 − 0.563i)19-s + (−0.673 + 0.739i)21-s − 1.02i·23-s + 0.764·25-s + (−0.406 + 0.913i)27-s + (0.0442 + 0.0255i)29-s + (−0.249 − 0.143i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.593801 - 0.397355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593801 - 0.397355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.60 - 0.649i)T \) |
| 7 | \( 1 + (-2.38 + 1.14i)T \) |
good | 5 | \( 1 + 2.96T + 5T^{2} \) |
| 11 | \( 1 + 4.72iT - 11T^{2} \) |
| 13 | \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.835 - 1.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.25 + 2.45i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.91iT - 23T^{2} \) |
| 29 | \( 1 + (-0.238 - 0.137i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.38 + 0.801i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.49 + 9.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.707 - 0.408i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.37 + 2.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.23 - 3.60i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (-13.6 + 7.88i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.15 - 10.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 6.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.60 - 7.98i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.00 + 4.04i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.55515576196800596752027556142, −11.01005795928004657753088320333, −10.49344358060274901104420082383, −8.653224586983970219303154348645, −8.099927914216843641194906961146, −6.79860217323318566439958785802, −5.66794034663862783932071507822, −4.41999664402837367329464049164, −3.59185170081974633967349128685, −0.68510219330231687961062531812,
1.72115399198710913501927001541, 4.05353445818109786354387418646, 4.87159499693056306293360807039, 6.21136074769236182264447630851, 7.43800938459405721340937693223, 7.964273703664534441805470895565, 9.310147486998931259361856740409, 10.77805280763496718586779141906, 11.33744173462191866188202491852, 12.16218711593994932866873869923