| L(s) = 1 | + (−20.3 − 11.7i)2-s + (−234. − 406. i)4-s + (−3.83e3 + 2.21e3i)5-s + (1.07e4 − 1.86e4i)7-s + 3.51e4i·8-s + 1.04e5·10-s + (−2.10e5 − 1.21e5i)11-s + (1.99e5 + 3.44e5i)13-s + (−4.39e5 + 2.53e5i)14-s + (1.73e5 − 3.00e5i)16-s + 2.02e6i·17-s − 2.77e6·19-s + (1.79e6 + 1.03e6i)20-s + (2.85e6 + 4.95e6i)22-s + (−2.14e6 + 1.23e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.367i)2-s + (−0.229 − 0.397i)4-s + (−1.22 + 0.707i)5-s + (0.641 − 1.11i)7-s + 1.07i·8-s + 1.04·10-s + (−1.30 − 0.753i)11-s + (0.536 + 0.928i)13-s + (−0.817 + 0.472i)14-s + (0.165 − 0.286i)16-s + 1.42i·17-s − 1.11·19-s + (0.562 + 0.324i)20-s + (0.554 + 0.960i)22-s + (−0.333 + 0.192i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.453088 - 0.317256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.453088 - 0.317256i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (20.3 + 11.7i)T + (512 + 886. i)T^{2} \) |
| 5 | \( 1 + (3.83e3 - 2.21e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-1.07e4 + 1.86e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (2.10e5 + 1.21e5i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-1.99e5 - 3.44e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 2.02e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 2.77e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (2.14e6 - 1.23e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-7.90e6 - 4.56e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-9.07e6 - 1.57e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 1.14e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.18e8 - 6.85e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (4.93e7 - 8.55e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (1.22e8 + 7.09e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 1.39e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-9.24e8 + 5.33e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-7.24e8 + 1.25e9i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (7.23e8 + 1.25e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.33e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.46e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-2.47e9 + 4.29e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (4.85e8 + 2.80e8i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + 4.73e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (6.08e9 - 1.05e10i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−11.52497726240081677688415793831, −10.81779029711110319352800783529, −10.32912042940994169233873548653, −8.423233803830936354078492773697, −7.958291650030789443803648040011, −6.46168388573289327321565110310, −4.69448309122513171040056053618, −3.55037730530316578161733530815, −1.76172614879010536787154715094, −0.36102635367082589484062159624,
0.56342235063319818280436791010, 2.62438791710893650431194997036, 4.26063501502519282918951844822, 5.31200522295194294018613764977, 7.28755872801906348748939824101, 8.249878214601085217154183457968, 8.617174077215711590429194118886, 10.10580415823092027660900202789, 11.62707089926845934672389547001, 12.42174243900286374978156892973
