| L(s) = 1 | + (−0.137 + 1.40i)2-s + (2.58 − 1.38i)3-s + (−1.96 − 0.387i)4-s + (0.634 + 0.773i)5-s + (1.59 + 3.83i)6-s + (−3.47 + 2.32i)7-s + (0.814 − 2.70i)8-s + (3.11 − 4.65i)9-s + (−1.17 + 0.786i)10-s + (1.86 − 0.565i)11-s + (−5.61 + 1.71i)12-s + (4.48 + 3.67i)13-s + (−2.79 − 5.21i)14-s + (2.70 + 1.12i)15-s + (3.70 + 1.51i)16-s + (6.31 − 2.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.0972 + 0.995i)2-s + (1.49 − 0.798i)3-s + (−0.981 − 0.193i)4-s + (0.283 + 0.345i)5-s + (0.649 + 1.56i)6-s + (−1.31 + 0.877i)7-s + (0.288 − 0.957i)8-s + (1.03 − 1.55i)9-s + (−0.371 + 0.248i)10-s + (0.561 − 0.170i)11-s + (−1.61 + 0.494i)12-s + (1.24 + 1.02i)13-s + (−0.745 − 1.39i)14-s + (0.699 + 0.289i)15-s + (0.925 + 0.379i)16-s + (1.53 − 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84113 + 1.04965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84113 + 1.04965i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.137 - 1.40i)T \) |
| 5 | \( 1 + (-0.634 - 0.773i)T \) |
| good | 3 | \( 1 + (-2.58 + 1.38i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (3.47 - 2.32i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.86 + 0.565i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-4.48 - 3.67i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-6.31 + 2.61i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.566 - 5.74i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (0.506 - 0.100i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.871 + 2.87i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (2.95 + 2.95i)T + 31iT^{2} \) |
| 37 | \( 1 + (7.66 + 0.754i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.03 + 5.22i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.61 - 1.39i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-4.61 - 11.1i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (3.70 + 12.2i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (5.81 - 4.77i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (1.61 + 3.01i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (0.780 + 1.46i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.25 + 6.37i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (11.0 + 7.40i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.14 + 5.18i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.506 - 0.0498i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (7.58 + 1.50i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (4.22 + 4.22i)T + 97iT^{2} \) |
| 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
−10.15428721090754209620982095969, −9.290265754880451293271245166701, −9.035281525545161359635868555284, −8.045304793915462797163417912663, −7.24548137615246599619559026694, −6.32017619013586578281450801046, −5.84586044014653753239326963927, −3.77649907232151594660584539847, −3.19340286196988837920763640080, −1.60026315016116703925066423038,
1.25771013883961059582540664425, 3.02624386873136578260431343858, 3.44690966324341542801792578813, 4.25972232952637607418016181697, 5.60770675319405190012375994160, 7.18030549487981154278468929580, 8.314072946428769932106218082012, 8.919762859633975096753505666957, 9.654848495637721174922539403363, 10.27270516173671789229494748706
