L(s) = 1 | + (−1.85 − 0.756i)2-s + (2.85 + 2.80i)4-s + (−3.70 − 6.41i)5-s + (−8.20 − 4.73i)7-s + (−3.16 − 7.34i)8-s + (2.00 + 14.6i)10-s + (−5.86 − 3.38i)11-s + (7.20 + 12.4i)13-s + (11.6 + 14.9i)14-s + (0.291 + 15.9i)16-s + 17.8·17-s − 5.19i·19-s + (7.40 − 28.6i)20-s + (8.29 + 10.7i)22-s + (−21.5 + 12.4i)23-s + ⋯ |
L(s) = 1 | + (−0.925 − 0.378i)2-s + (0.713 + 0.700i)4-s + (−0.740 − 1.28i)5-s + (−1.17 − 0.677i)7-s + (−0.395 − 0.918i)8-s + (0.200 + 1.46i)10-s + (−0.533 − 0.307i)11-s + (0.554 + 0.960i)13-s + (0.829 + 1.07i)14-s + (0.0182 + 0.999i)16-s + 1.05·17-s − 0.273i·19-s + (0.370 − 1.43i)20-s + (0.376 + 0.486i)22-s + (−0.938 + 0.542i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0962 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0962 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0603025 + 0.0664140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0603025 + 0.0664140i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.85 + 0.756i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (3.70 + 6.41i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (8.20 + 4.73i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (5.86 + 3.38i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.20 - 12.4i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 17.8T + 289T^{2} \) |
| 19 | \( 1 + 5.19iT - 361T^{2} \) |
| 23 | \( 1 + (21.5 - 12.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (14.8 - 25.6i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (14.8 - 8.56i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 6.41T + 1.36e3T^{2} \) |
| 41 | \( 1 + (4.32 + 7.48i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-43.4 - 25.0i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-45.0 - 26.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 82.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (72.5 - 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.20 + 7.28i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-25.1 + 14.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 2.16T + 5.32e3T^{2} \) |
| 79 | \( 1 + (129. + 74.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-11.7 - 6.77i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 17.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-69.1 + 119. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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
−11.58672326067226762475645886246, −10.64369047664825491365875440177, −9.599621499056693639031491160070, −8.978111221843612326204142467205, −7.977783552567673902860595972215, −7.19164920980061052204419610656, −5.90267184071140402996924871132, −4.21375140630099481587966613009, −3.29403652917845570625225828745, −1.25072959117205332606473485477,
0.06192059298469976378578384720, 2.54130157857475170490874410089, 3.49610183042715645092110285688, 5.71367038169143565577131803436, 6.35938092581910296879907788641, 7.51448709514521354749489073258, 8.054917358867819425197552305627, 9.391712833597673278780382409470, 10.22054873438389851709960780246, 10.81573902496451089366607553206