L(s) = 1 | + (−4.62 + 4.62i)5-s + 3.04·7-s + (−9.15 − 9.15i)11-s + (5.78 + 5.78i)13-s − 17.6·17-s + (1.15 − 1.15i)19-s + 3.45·23-s − 17.8i·25-s + (12.1 + 12.1i)29-s − 38.5i·31-s + (−14.1 + 14.1i)35-s + (0.0972 − 0.0972i)37-s − 51.5i·41-s + (1.70 + 1.70i)43-s + 24.1i·47-s + ⋯ |
L(s) = 1 | + (−0.925 + 0.925i)5-s + 0.435·7-s + (−0.831 − 0.831i)11-s + (0.444 + 0.444i)13-s − 1.03·17-s + (0.0606 − 0.0606i)19-s + 0.150·23-s − 0.712i·25-s + (0.420 + 0.420i)29-s − 1.24i·31-s + (−0.403 + 0.403i)35-s + (0.00262 − 0.00262i)37-s − 1.25i·41-s + (0.0395 + 0.0395i)43-s + 0.513i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.009958349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009958349\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (4.62 - 4.62i)T - 25iT^{2} \) |
| 7 | \( 1 - 3.04T + 49T^{2} \) |
| 11 | \( 1 + (9.15 + 9.15i)T + 121iT^{2} \) |
| 13 | \( 1 + (-5.78 - 5.78i)T + 169iT^{2} \) |
| 17 | \( 1 + 17.6T + 289T^{2} \) |
| 19 | \( 1 + (-1.15 + 1.15i)T - 361iT^{2} \) |
| 23 | \( 1 - 3.45T + 529T^{2} \) |
| 29 | \( 1 + (-12.1 - 12.1i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.5iT - 961T^{2} \) |
| 37 | \( 1 + (-0.0972 + 0.0972i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-1.70 - 1.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 24.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.0 + 27.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-19.5 - 19.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (16.7 + 16.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-75.8 + 75.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 135. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-74.9 + 74.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 31.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.5T + 9.40e3T^{2} \) |
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
−9.410959417061366609816513580772, −8.454922603463098899806051458488, −7.87433788043988228808860109466, −7.01290428845593125485332158201, −6.24654063281099088129885281486, −5.13464615929505603251419125248, −4.08577818189622806826178611343, −3.24346188490794340159391436261, −2.17895952697658352631442953173, −0.36582841150529741602320954291,
0.995725447331528654900335356291, 2.37888539854800271455791720371, 3.72798114466357988032424353460, 4.70298628808896434415955930990, 5.14163586622919933671770061010, 6.48223820780125026814592998715, 7.45920916302465408236508039481, 8.220494059674924703124156566286, 8.659884266093022666151045381270, 9.724865772975707699362588276817