L(s) = 1 | + (−1.97 − 0.323i)2-s + (1.70 + 2.95i)3-s + (3.79 + 1.27i)4-s + (−1.34 − 0.774i)5-s + (−2.41 − 6.38i)6-s + (−7.07 − 3.74i)8-s + (−1.32 + 2.30i)9-s + (2.39 + 1.96i)10-s + (2.24 + 3.88i)11-s + (2.70 + 13.3i)12-s − 1.54i·13-s − 5.29i·15-s + (12.7 + 9.66i)16-s + (11.8 + 20.4i)17-s + (3.36 − 4.11i)18-s + (−12.4 + 21.5i)19-s + ⋯ |
L(s) = 1 | + (−0.986 − 0.161i)2-s + (0.569 + 0.985i)3-s + (0.947 + 0.318i)4-s + (−0.268 − 0.154i)5-s + (−0.402 − 1.06i)6-s + (−0.883 − 0.467i)8-s + (−0.147 + 0.255i)9-s + (0.239 + 0.196i)10-s + (0.203 + 0.353i)11-s + (0.225 + 1.11i)12-s − 0.119i·13-s − 0.352i·15-s + (0.796 + 0.604i)16-s + (0.695 + 1.20i)17-s + (0.186 − 0.228i)18-s + (−0.654 + 1.13i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.662521 + 0.889471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.662521 + 0.889471i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.97 + 0.323i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.70 - 2.95i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (1.34 + 0.774i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.24 - 3.88i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 1.54iT - 169T^{2} \) |
| 17 | \( 1 + (-11.8 - 20.4i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.4 - 21.5i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-30.5 - 17.6i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 22.4iT - 841T^{2} \) |
| 31 | \( 1 + (40.4 - 23.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (50.7 + 29.2i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 17.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-31.2 - 18.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (84.7 - 48.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-30.7 - 53.3i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-32.6 - 18.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.6 - 28.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-34.6 - 60.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-33.5 - 19.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 3.61T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-22.0 + 38.1i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 96.1T + 9.40e3T^{2} \) |
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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
−10.83447007182672324360570191051, −10.41532597869834545521051741755, −9.459423343375504078315364045526, −8.795100783867377791589373163062, −7.982296953528682148722962637753, −6.93464231234156808592954282017, −5.62312801357325811393005443804, −4.03418178907144871840195362107, −3.26528809515673103005807961746, −1.58105303880056688812369512005,
0.64136042506000896168684520372, 2.09122054490692145395137687905, 3.17179893582484471589361714285, 5.17018245017311139803484951041, 6.63785586731138234082859146871, 7.17862354294477861587133177623, 8.003938288553907538955542276501, 8.857752922443501698914231436222, 9.629595551408923608122442292643, 10.91110450249774552189886355469