L(s) = 1 | + 6.80i·2-s + (−12.4 − 9.40i)3-s − 14.2·4-s − 14.9·5-s + (63.9 − 84.5i)6-s + 120. i·8-s + (66.1 + 233. i)9-s − 101. i·10-s + 96.0i·11-s + (177. + 134. i)12-s + 416. i·13-s + (185. + 140. i)15-s − 1.27e3·16-s − 208.·17-s + (−1.59e3 + 449. i)18-s − 2.65e3i·19-s + ⋯ |
L(s) = 1 | + 1.20i·2-s + (−0.797 − 0.603i)3-s − 0.446·4-s − 0.266·5-s + (0.725 − 0.959i)6-s + 0.665i·8-s + (0.272 + 0.962i)9-s − 0.320i·10-s + 0.239i·11-s + (0.356 + 0.269i)12-s + 0.684i·13-s + (0.212 + 0.160i)15-s − 1.24·16-s − 0.174·17-s + (−1.15 + 0.327i)18-s − 1.68i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2781670510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2781670510\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (12.4 + 9.40i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 6.80iT - 32T^{2} \) |
| 5 | \( 1 + 14.9T + 3.12e3T^{2} \) |
| 11 | \( 1 - 96.0iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 416. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 208.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.65e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.61e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.22e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.29e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 3.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.62e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 7.82e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.01e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.60e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 5.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.51e4iT - 8.58e9T^{2} \) |
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
−11.56714035106444438499121283772, −11.45012078619266426608904792882, −9.723227047853093270629934918988, −8.328055554109727416740707630771, −7.35137489751888486313103052236, −6.64489365117227439393410852131, −5.60955019652951449148877639209, −4.50389614627183136387039370909, −2.10562879603227127425838345521, −0.10725954061039257545504679423,
1.33631604950505413491778201367, 3.17488305792809675552282060136, 4.16707630999288839736689632069, 5.58172108954014134161454131927, 6.85089787031665747108906635259, 8.488817934476491025051465946276, 9.869066502203482190758176966108, 10.46865664878581834600611945155, 11.27174026200751207009601206783, 12.23288516978045370466748494553