L(s) = 1 | + (−2.35 + 0.631i)2-s + (1.54 + 0.775i)3-s + (3.42 − 1.97i)4-s + (−4.13 − 0.849i)6-s + (1.91 + 1.82i)7-s + (−3.36 + 3.36i)8-s + (1.79 + 2.40i)9-s + (−3.08 + 1.77i)11-s + (6.83 − 0.406i)12-s + (−1.28 − 1.28i)13-s + (−5.67 − 3.08i)14-s + (1.85 − 3.21i)16-s + (−0.792 + 2.95i)17-s + (−5.75 − 4.52i)18-s + (0.331 + 0.191i)19-s + ⋯ |
L(s) = 1 | + (−1.66 + 0.446i)2-s + (0.894 + 0.447i)3-s + (1.71 − 0.987i)4-s + (−1.68 − 0.346i)6-s + (0.725 + 0.688i)7-s + (−1.19 + 1.19i)8-s + (0.599 + 0.800i)9-s + (−0.928 + 0.536i)11-s + (1.97 − 0.117i)12-s + (−0.356 − 0.356i)13-s + (−1.51 − 0.823i)14-s + (0.463 − 0.803i)16-s + (−0.192 + 0.717i)17-s + (−1.35 − 1.06i)18-s + (0.0761 + 0.0439i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.454983 + 0.717563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.454983 + 0.717563i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.54 - 0.775i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
good | 2 | \( 1 + (2.35 - 0.631i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (3.08 - 1.77i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 + 1.28i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.792 - 2.95i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.331 - 0.191i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.658 - 2.45i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + (-0.323 - 0.561i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.34 - 5.00i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (-0.335 - 0.335i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.80 - 0.751i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.04 - 0.815i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.81 - 6.60i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.45 - 9.45i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.3 + 3.31i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (-0.849 + 3.17i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.21 + 1.85i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 + 0.973i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.51 + 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \) |
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
−10.46016648065630938621692700571, −10.22760512689162687324219696993, −9.130474406087089829820676151383, −8.542167681261378540072115255108, −7.84584983113035563048593191473, −7.20228758962265055645400019454, −5.75762447488437032878779130023, −4.60950923747197738846478569319, −2.75575311934874522123456836811, −1.73914002376074862184833152357,
0.78185260655787997983374329136, 2.11821973974800748685113959421, 3.08350223428749731434773069313, 4.69736162312615691591435310681, 6.60247259110152183501889362352, 7.48077373468627034976785473193, 8.039093506169542786578755609003, 8.729853538738128235069675160254, 9.622389135019257372325424581849, 10.36840478366603478637397848295