L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.965 + 0.258i)3-s + (−0.499 − 0.866i)4-s + (−1.90 + 1.17i)5-s + (−0.258 + 0.965i)6-s + (3.41 − 1.97i)7-s − 0.999·8-s + (0.866 − 0.499i)9-s + (0.0614 + 2.23i)10-s + (−1.48 − 5.56i)11-s + (0.707 + 0.707i)12-s + (−3.53 + 0.730i)13-s − 3.94i·14-s + (1.53 − 1.62i)15-s + (−0.5 + 0.866i)16-s + (1.03 − 3.85i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.557 + 0.149i)3-s + (−0.249 − 0.433i)4-s + (−0.851 + 0.523i)5-s + (−0.105 + 0.394i)6-s + (1.29 − 0.745i)7-s − 0.353·8-s + (0.288 − 0.166i)9-s + (0.0194 + 0.706i)10-s + (−0.449 − 1.67i)11-s + (0.204 + 0.204i)12-s + (−0.979 + 0.202i)13-s − 1.05i·14-s + (0.396 − 0.419i)15-s + (−0.125 + 0.216i)16-s + (0.250 − 0.936i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.502969 - 0.906860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.502969 - 0.906860i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (1.90 - 1.17i)T \) |
| 13 | \( 1 + (3.53 - 0.730i)T \) |
good | 7 | \( 1 + (-3.41 + 1.97i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.48 + 5.56i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.03 + 3.85i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 0.558i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (1.12 + 4.18i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (5.04 + 2.91i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.83 + 2.83i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.16 - 1.82i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0577 - 0.0154i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-10.3 - 2.76i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 13.4iT - 47T^{2} \) |
| 53 | \( 1 + (-4.15 - 4.15i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.509 - 1.89i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.36 - 2.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.31 + 12.3i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 + 2.49iT - 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 + (-8.00 + 2.14i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.76 + 8.26i)T + (-48.5 + 84.0i)T^{2} \) |
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
−11.11020661001340585613017075690, −10.62537873725821229191028129878, −9.396440064016212370029139599649, −7.995934227458129084065521148925, −7.43139630649731872525550402935, −5.97621501566107294944158128636, −4.88191230080525495553939945928, −4.04965434598519014694110773113, −2.75734912371833160054119553839, −0.67366813922846069908697321924,
1.96054834118812882202553675996, 4.04008418617730015531498056840, 5.07582166503487166849711959959, 5.43562416885712796039963682266, 7.26673879150688290133439050290, 7.59034390528736870423941239948, 8.593976021224621698600827987699, 9.711868444420035191266859983798, 10.96150295203955388090268747677, 11.95776008136910604004057822551