L(s) = 1 | + (−1.02 + 0.520i)2-s + (−1.04 − 1.38i)3-s + (−0.404 + 0.556i)4-s + (1.78 + 0.864i)6-s + (−4.96 + 0.785i)7-s + (0.481 − 3.04i)8-s + (−0.810 + 2.88i)9-s + (−2.47 + 2.20i)11-s + (1.19 − 0.0241i)12-s + (1.35 − 0.692i)13-s + (4.65 − 3.38i)14-s + (0.664 + 2.04i)16-s + (−0.498 + 0.979i)17-s + (−0.674 − 3.36i)18-s + (−2.83 − 3.90i)19-s + ⋯ |
L(s) = 1 | + (−0.721 + 0.367i)2-s + (−0.604 − 0.796i)3-s + (−0.202 + 0.278i)4-s + (0.729 + 0.353i)6-s + (−1.87 + 0.296i)7-s + (0.170 − 1.07i)8-s + (−0.270 + 0.962i)9-s + (−0.746 + 0.665i)11-s + (0.343 − 0.00697i)12-s + (0.377 − 0.192i)13-s + (1.24 − 0.903i)14-s + (0.166 + 0.511i)16-s + (−0.120 + 0.237i)17-s + (−0.159 − 0.794i)18-s + (−0.651 − 0.896i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.352184 - 0.0507865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.352184 - 0.0507865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.04 + 1.38i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (2.47 - 2.20i)T \) |
good | 2 | \( 1 + (1.02 - 0.520i)T + (1.17 - 1.61i)T^{2} \) |
| 7 | \( 1 + (4.96 - 0.785i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-1.35 + 0.692i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.498 - 0.979i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (2.83 + 3.90i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.45 - 2.45i)T - 23iT^{2} \) |
| 29 | \( 1 + (-6.36 - 4.62i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.04 + 3.20i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.51 + 0.398i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (1.70 + 2.35i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (6.92 + 6.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.43 - 0.860i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (1.21 + 2.37i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (3.38 + 2.45i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.115 - 0.354i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.65 - 1.65i)T - 67iT^{2} \) |
| 71 | \( 1 + (-12.2 + 3.97i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.481 + 3.04i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 3.28i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-13.9 - 7.12i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + (-4.17 - 8.19i)T + (-57.0 + 78.4i)T^{2} \) |
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
−10.10072679263859842778061983941, −9.271004930620075729407997596456, −8.444436958071184644162954498968, −7.54772806509781149909829347875, −6.71711280870689104136555640681, −6.28753426112915543928167626677, −5.04170450296181584245004843625, −3.62999082821569766805496736533, −2.44466106999584165879130254278, −0.44513989555801328115846072185,
0.63064412024308969190415163329, 2.75105539699993182489384635995, 3.79302586310743734595430643622, 4.90304804485625802010665928704, 6.12501633091431741553663098986, 6.37502843348656772054771789430, 8.035155128525821401229184793130, 8.905970912899255273254059837816, 9.650405314474177987133585780709, 10.31715594084736309775699610773