| L(s) = 1 | + (−1.37 + 2.37i)2-s + (−4.5 − 7.79i)3-s + (12.2 + 21.1i)4-s + (−29.1 + 50.5i)5-s + 24.7·6-s + (−21.4 + 127. i)7-s − 155.·8-s + (−40.5 + 70.1i)9-s + (−80.2 − 138. i)10-s + (−8.71 − 15.0i)11-s + (110. − 190. i)12-s + 889.·13-s + (−274. − 226. i)14-s + 525.·15-s + (−178. + 308. i)16-s + (−513. − 889. i)17-s + ⋯ |
| L(s) = 1 | + (−0.242 + 0.420i)2-s + (−0.288 − 0.499i)3-s + (0.381 + 0.661i)4-s + (−0.522 + 0.904i)5-s + 0.280·6-s + (−0.165 + 0.986i)7-s − 0.856·8-s + (−0.166 + 0.288i)9-s + (−0.253 − 0.439i)10-s + (−0.0217 − 0.0376i)11-s + (0.220 − 0.381i)12-s + 1.46·13-s + (−0.374 − 0.309i)14-s + 0.602·15-s + (−0.173 + 0.301i)16-s + (−0.430 − 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.545285 + 0.841488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.545285 + 0.841488i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 + (21.4 - 127. i)T \) |
| good | 2 | \( 1 + (1.37 - 2.37i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (29.1 - 50.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (8.71 + 15.0i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 889.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (513. + 889. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-869. + 1.50e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.96e3 - 3.40e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 5.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.54e3 - 2.68e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.51e3 - 4.35e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.83e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.80e3 + 8.31e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.16e4 - 2.01e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.80e3 - 3.12e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.14e4 - 1.98e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.35e4 + 4.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.96e3 + 5.13e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.42e4 + 7.66e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 9.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.32e4 + 4.02e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 7.59e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.81574739317051956140359181110, −15.98101743577258015112160175777, −15.46644348595637589256487844052, −13.59455268409876355724893815762, −11.99422334879709212948415596930, −11.20038925545186820433850765906, −8.815446820139373772497702640919, −7.38864140114291561487345493920, −6.17114696351992462762313902527, −3.00472761016487634143420299192,
0.863360223626273806926024241837, 4.14103377745463855692439433648, 6.16897695149997822842831244767, 8.471450087768290611909310793405, 10.10941390142948476807388514577, 11.07906418964626679326663675533, 12.47843423998954456470072486075, 14.18662006216146572729670260195, 15.81245868334154304808522791770, 16.42927533690574202639852672066
