| L(s) = 1 | + (39 + 67.5i)2-s + (−2.01e3 + 3.49e3i)4-s + (−2.68e3 + 4.65e3i)5-s + (1.38e4 + 2.40e4i)7-s − 1.55e5·8-s − 4.18e5·10-s + (3.18e5 + 5.52e5i)11-s + (−3.83e5 + 6.63e5i)13-s + (−1.08e6 + 1.87e6i)14-s + (−1.91e6 − 3.31e6i)16-s − 3.08e6·17-s − 1.95e7·19-s + (−1.08e7 − 1.87e7i)20-s + (−2.48e7 + 4.30e7i)22-s + (7.65e6 − 1.32e7i)23-s + ⋯ |
| L(s) = 1 | + (0.861 + 1.49i)2-s + (−0.985 + 1.70i)4-s + (−0.384 + 0.665i)5-s + (0.312 + 0.540i)7-s − 1.67·8-s − 1.32·10-s + (0.597 + 1.03i)11-s + (−0.286 + 0.495i)13-s + (−0.537 + 0.931i)14-s + (−0.456 − 0.790i)16-s − 0.526·17-s − 1.80·19-s + (−0.757 − 1.31i)20-s + (−1.02 + 1.78i)22-s + (0.248 − 0.429i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.53246 - 1.28588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53246 - 1.28588i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-39 - 67.5i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (2.68e3 - 4.65e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-1.38e4 - 2.40e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-3.18e5 - 5.52e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (3.83e5 - 6.63e5i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 + 3.08e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.95e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-7.65e6 + 1.32e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-5.37e6 - 9.31e6i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-2.54e7 + 4.41e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 - 6.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-4.49e8 + 7.78e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-4.78e8 - 8.29e8i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (7.77e8 + 1.34e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + 3.79e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-2.77e8 + 4.80e8i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.47e9 + 4.28e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.64e9 - 4.58e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.48e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.39e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.86e9 + 3.22e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-4.38e9 - 7.59e9i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 - 2.54e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.95e10 - 3.38e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−13.11215664602385107194272977196, −12.24899929254597359429373322056, −10.97022500519927352512357603502, −9.214647357935758646938249326612, −8.062697900965566829609063336082, −6.95191229657837817845357266693, −6.29843651556212874023769557961, −4.79888170807912448746529478546, −3.99655996671882731014820617686, −2.26463857484741169562861023511,
0.38116024685868552508610705155, 1.27372012588068908729071916165, 2.69818216519492721293403845827, 4.01368461311301499920948955531, 4.68955135337533552514672721750, 6.15498479276903752156961097800, 8.124242050128200712634844298775, 9.310115664869278777501182440321, 10.69896395328426401811706637753, 11.25156076254746166998604602883
