L(s) = 1 | + (8.95 − 5.16i)2-s + (1.78 + 15.4i)3-s + (37.4 − 64.7i)4-s + (−15.8 − 27.4i)5-s + (95.9 + 129. i)6-s + (20.8 + 127. i)7-s − 442. i·8-s + (−236. + 55.1i)9-s + (−284. − 164. i)10-s + (−392. − 226. i)11-s + (1.06e3 + 463. i)12-s + 551. i·13-s + (847. + 1.03e3i)14-s + (397. − 294. i)15-s + (−1.08e3 − 1.88e3i)16-s + (−269. + 466. i)17-s + ⋯ |
L(s) = 1 | + (1.58 − 0.913i)2-s + (0.114 + 0.993i)3-s + (1.16 − 2.02i)4-s + (−0.283 − 0.491i)5-s + (1.08 + 1.46i)6-s + (0.160 + 0.986i)7-s − 2.44i·8-s + (−0.973 + 0.227i)9-s + (−0.898 − 0.518i)10-s + (−0.978 − 0.565i)11-s + (2.14 + 0.929i)12-s + 0.905i·13-s + (1.15 + 1.41i)14-s + (0.456 − 0.338i)15-s + (−1.06 − 1.84i)16-s + (−0.225 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.77281 - 0.925763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.77281 - 0.925763i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.78 - 15.4i)T \) |
| 7 | \( 1 + (-20.8 - 127. i)T \) |
good | 2 | \( 1 + (-8.95 + 5.16i)T + (16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (15.8 + 27.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (392. + 226. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 551. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (269. - 466. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.18e3 + 683. i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.80e3 + 1.61e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 1.60e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (6.12e3 + 3.53e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-975. - 1.68e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.04e3 - 5.27e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.22e4 - 7.08e3i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (8.45e3 - 1.46e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.57e4 - 1.48e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.80e3 - 1.17e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.13e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-8.50e3 - 4.91e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.98e4 + 8.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (476. + 825. i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.15e5iT - 8.58e9T^{2} \) |
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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
−16.30108973701545920942545103013, −15.37374552322167825861536889100, −14.35165415333060603757602166023, −12.99095489242644766670514762555, −11.71794290233785451049030991703, −10.74294917092093630630969073338, −8.984152815154307635747939255675, −5.64030789550795605381540018431, −4.52678856753803353598223528240, −2.77267431365534771092584099666,
3.19643769795101262787324513717, 5.33070082186838628167889392355, 7.12143698374622380377863037094, 7.67644551796847900855451468921, 11.10812630485903903187238578839, 12.63561258667160395090095188009, 13.43683090083959913535627437602, 14.48448468061693171461809106872, 15.54970085356923964416695187987, 17.01541508677998425745760068921