| L(s) = 1 | + (−1.41 − 0.0602i)2-s + (0.453 + 0.891i)3-s + (1.99 + 0.170i)4-s + (−1.79 + 1.33i)5-s + (−0.587 − 1.28i)6-s + (−2.79 − 2.79i)7-s + (−2.80 − 0.360i)8-s + (−0.587 + 0.809i)9-s + (2.61 − 1.78i)10-s + (−1.49 − 2.06i)11-s + (0.752 + 1.85i)12-s + (−1.10 − 6.98i)13-s + (3.78 + 4.11i)14-s + (−2.00 − 0.989i)15-s + (3.94 + 0.679i)16-s + (2.15 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0426i)2-s + (0.262 + 0.514i)3-s + (0.996 + 0.0851i)4-s + (−0.801 + 0.598i)5-s + (−0.239 − 0.525i)6-s + (−1.05 − 1.05i)7-s + (−0.991 − 0.127i)8-s + (−0.195 + 0.269i)9-s + (0.826 − 0.563i)10-s + (−0.451 − 0.621i)11-s + (0.217 + 0.534i)12-s + (−0.306 − 1.93i)13-s + (1.01 + 1.10i)14-s + (−0.517 − 0.255i)15-s + (0.985 + 0.169i)16-s + (0.522 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.176591 - 0.267887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.176591 - 0.267887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0602i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (1.79 - 1.33i)T \) |
| good | 7 | \( 1 + (2.79 + 2.79i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.49 + 2.06i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.10 + 6.98i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.15 - 1.09i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 3.63i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.422 + 2.66i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (3.49 + 1.13i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.552 + 0.179i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.38 - 1.01i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (9.43 + 6.85i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.64 - 2.64i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.64 - 1.85i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 0.757i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (3.32 + 2.41i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.266 + 0.193i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.22 - 12.2i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-13.4 - 4.38i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.3 - 1.63i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.99 + 6.13i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (11.3 + 5.79i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.98 + 8.24i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0710 - 0.139i)T + (-57.0 + 78.4i)T^{2} \) |
| 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
−10.96931061014458998449404202825, −10.26358764628403848195513215121, −10.00203384149950292719551465552, −8.411650824166604783743288144341, −7.80077235315400100516958004441, −6.90233380040962318015103361156, −5.64544479891640393594955487776, −3.56046245171834016730876587645, −3.08742899558272797762229568343, −0.29671261753998095749478854834,
1.92808630135504100557015742544, 3.29621854061960916982513310562, 5.15582608433981723385239534103, 6.62678823135298912282032850446, 7.24436335617140433803245523341, 8.378474393402125363329496334987, 9.243292632854862553638911587248, 9.648819230405284215818199622597, 11.30327158958651458849136412553, 12.03459399708340945042536767213
