| L(s) = 1 | + (0.982 − 0.473i)2-s + (−0.504 + 0.633i)4-s + (−1.29 + 0.398i)5-s + (−0.108 + 0.187i)7-s + (−0.682 + 2.98i)8-s + (−1.08 + 1.00i)10-s + (3.76 + 4.71i)11-s + (2.10 + 1.95i)13-s + (−0.0176 + 0.235i)14-s + (0.383 + 1.68i)16-s + (−0.270 − 0.0833i)17-s + (1.12 + 0.169i)19-s + (0.399 − 1.01i)20-s + (5.92 + 2.85i)22-s + (1.44 − 3.67i)23-s + ⋯ |
| L(s) = 1 | + (0.695 − 0.334i)2-s + (−0.252 + 0.316i)4-s + (−0.577 + 0.178i)5-s + (−0.0408 + 0.0708i)7-s + (−0.241 + 1.05i)8-s + (−0.341 + 0.316i)10-s + (1.13 + 1.42i)11-s + (0.584 + 0.542i)13-s + (−0.00471 + 0.0629i)14-s + (0.0959 + 0.420i)16-s + (−0.0654 − 0.0202i)17-s + (0.257 + 0.0388i)19-s + (0.0893 − 0.227i)20-s + (1.26 + 0.608i)22-s + (0.300 − 0.765i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37656 + 0.749187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37656 + 0.749187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-5.90 - 2.85i)T \) |
| good | 2 | \( 1 + (-0.982 + 0.473i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (1.29 - 0.398i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.108 - 0.187i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.76 - 4.71i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 1.95i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.270 + 0.0833i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 0.169i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (-1.44 + 3.67i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (0.515 - 6.88i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (8.17 + 5.57i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (3.77 + 6.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.62 + 2.22i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (0.288 - 0.361i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-6.12 + 5.68i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 8.14i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-11.0 + 7.56i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-6.22 - 0.938i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (-0.540 - 1.37i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-0.601 - 0.557i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (3.07 - 5.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.538 - 7.18i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (0.331 + 4.42i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (2.78 + 3.49i)T + (-21.5 + 94.5i)T^{2} \) |
| 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
−11.55963582027742003031854535456, −10.98016752066244531856731509897, −9.479258813281198641587742912754, −8.865587470728399943210154831823, −7.62441206073476710704881697301, −6.81165209715118091671982248131, −5.41475894605652390947308977630, −4.20600725048366014477250486283, −3.69777827429677714218992462948, −2.07251379295597658340707639222,
0.922655857009544204544903042382, 3.45472269538513218753788489943, 4.08295188111308937653375040382, 5.48174635563723001437311027357, 6.12963717311170044164838644913, 7.25337830033593768694346211541, 8.495405720369929517868854140107, 9.194889598336060701254308269907, 10.34366410741665176260686298663, 11.35682513197701111389211949262
