| 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
−17.01541508677998425745760068921, −15.54970085356923964416695187987, −14.48448468061693171461809106872, −13.43683090083959913535627437602, −12.63561258667160395090095188009, −11.10812630485903903187238578839, −7.67644551796847900855451468921, −7.12143698374622380377863037094, −5.33070082186838628167889392355, −3.19643769795101262787324513717,
2.77267431365534771092584099666, 4.52678856753803353598223528240, 5.64030789550795605381540018431, 8.984152815154307635747939255675, 10.74294917092093630630969073338, 11.71794290233785451049030991703, 12.99095489242644766670514762555, 14.35165415333060603757602166023, 15.37374552322167825861536889100, 16.30108973701545920942545103013
