| L(s) = 1 | + (3.23 − 2.35i)2-s − 13.3i·3-s + (4.94 − 15.2i)4-s + (−5.46 + 16.8i)5-s + (−31.4 − 43.2i)6-s + (93.7 − 129. i)7-s + (−19.7 − 60.8i)8-s + 64.5·9-s + (21.8 + 67.2i)10-s + (−207. + 67.4i)11-s + (−203. − 66.0i)12-s + (−646. − 889. i)13-s − 638. i·14-s + (224. + 73.0i)15-s + (−207. − 150. i)16-s + (−62.7 + 20.4i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s − 0.856i·3-s + (0.154 − 0.475i)4-s + (−0.0977 + 0.300i)5-s + (−0.356 − 0.490i)6-s + (0.723 − 0.995i)7-s + (−0.109 − 0.336i)8-s + 0.265·9-s + (0.0691 + 0.212i)10-s + (−0.517 + 0.168i)11-s + (−0.407 − 0.132i)12-s + (−1.06 − 1.46i)13-s − 0.870i·14-s + (0.257 + 0.0837i)15-s + (−0.202 − 0.146i)16-s + (−0.0526 + 0.0171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.700853 - 2.13235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.700853 - 2.13235i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 41 | \( 1 + (-8.21e3 + 6.95e3i)T \) |
| good | 3 | \( 1 + 13.3iT - 243T^{2} \) |
| 5 | \( 1 + (5.46 - 16.8i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-93.7 + 129. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (207. - 67.4i)T + (1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (646. + 889. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (62.7 - 20.4i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (361. - 497. i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (3.66e3 - 2.66e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-6.84e3 - 2.22e3i)T + (1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.07e3 + 9.47e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-797. + 2.45e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 43 | \( 1 + (-1.43e4 + 1.04e4i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-3.13e3 - 4.31e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.79e4 - 5.83e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.60e4 - 1.16e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.29e4 - 1.66e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-6.62e4 - 2.15e4i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.45e4 - 7.98e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + 4.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.73e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.05e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.29e4 + 1.77e4i)T + (-1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-5.12e4 - 1.66e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
| 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
−12.90542529510057923901896703927, −12.10293346087269532197958589804, −10.76930144076151581123378953075, −10.03036762438541540637747601839, −7.81408815568950776278152469358, −7.28344435116719736076846952426, −5.60277553075278463163416537011, −4.13911975535605933601454848322, −2.36016312092095020160852543613, −0.795062568572660260040820440957,
2.37459874096858696612052116359, 4.40673782118033472353950943411, 5.00134674629044908417716535988, 6.61009406999019656434317347611, 8.185420305434076671715622514846, 9.200988513301955018737180633133, 10.51364644517588463871458986987, 11.86587541094327046620023471553, 12.55621524577381484428843849648, 14.19037587014577729990154058573
