L(s) = 1 | + (0.667 − 1.15i)2-s + (−0.866 + 0.5i)3-s + (0.109 + 0.190i)4-s + (−0.786 − 0.453i)5-s + 1.33i·6-s + (−1.35 − 2.26i)7-s + 2.96·8-s + (0.499 − 0.866i)9-s + (−1.04 + 0.605i)10-s + (4.27 − 2.46i)11-s + (−0.190 − 0.109i)12-s + 1.97·13-s + (−3.52 + 0.0564i)14-s + 0.907·15-s + (1.75 − 3.04i)16-s + (−0.772 − 4.05i)17-s + ⋯ |
L(s) = 1 | + (0.471 − 0.817i)2-s + (−0.499 + 0.288i)3-s + (0.0549 + 0.0950i)4-s + (−0.351 − 0.203i)5-s + 0.544i·6-s + (−0.513 − 0.857i)7-s + 1.04·8-s + (0.166 − 0.288i)9-s + (−0.331 + 0.191i)10-s + (1.28 − 0.744i)11-s + (−0.0549 − 0.0316i)12-s + 0.546·13-s + (−0.943 + 0.0150i)14-s + 0.234·15-s + (0.439 − 0.760i)16-s + (−0.187 − 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22442 - 0.933563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22442 - 0.933563i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.35 + 2.26i)T \) |
| 17 | \( 1 + (0.772 + 4.05i)T \) |
good | 2 | \( 1 + (-0.667 + 1.15i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.786 + 0.453i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.27 + 2.46i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 19 | \( 1 + (0.0525 - 0.0910i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.49iT - 29T^{2} \) |
| 31 | \( 1 + (2.16 - 1.25i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.919 + 0.531i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.00iT - 41T^{2} \) |
| 43 | \( 1 + 9.82T + 43T^{2} \) |
| 47 | \( 1 + (-0.733 + 1.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.80 + 4.86i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.35 - 7.53i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.92 + 4.00i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.83 - 10.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.32iT - 71T^{2} \) |
| 73 | \( 1 + (0.228 - 0.131i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.84 + 1.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + (4.37 - 7.57i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.99iT - 97T^{2} \) |
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
−11.39160451690550244460120165374, −10.70336966769949016504059141864, −9.678995127558155344636305447214, −8.624812321966620638991607164443, −7.26930707520717791529518418746, −6.50145719684151696318789133196, −4.98744723523522428540469909200, −3.91327300170946386970882096788, −3.27167994487892869635824206213, −1.13407131161117361597612846389,
1.77362480896687087936176321974, 3.77737377401232542940792805175, 4.95702867623429117452682776463, 6.14064847081250075677412202130, 6.53710898062128258412725278824, 7.52699111654780341337759495084, 8.736035210368450542384435073783, 9.799268677926750325651702554126, 10.91379789078063370884722629500, 11.69975621991319853589209617512