| L(s) = 1 | + (25.4 + 44.1i)2-s + (−274. + 474. i)4-s + (−3.55e3 + 6.15e3i)5-s + (3.94e4 + 6.83e4i)7-s + 7.64e4·8-s − 3.61e5·10-s + (2.06e5 + 3.57e5i)11-s + (1.35e5 − 2.34e5i)13-s + (−2.01e6 + 3.48e6i)14-s + (2.50e6 + 4.34e6i)16-s + 1.90e6·17-s + 1.56e7·19-s + (−1.94e6 − 3.37e6i)20-s + (−1.05e7 + 1.82e7i)22-s + (2.13e7 − 3.69e7i)23-s + ⋯ |
| L(s) = 1 | + (0.562 + 0.975i)2-s + (−0.133 + 0.231i)4-s + (−0.508 + 0.880i)5-s + (0.887 + 1.53i)7-s + 0.824·8-s − 1.14·10-s + (0.386 + 0.668i)11-s + (0.101 − 0.175i)13-s + (−0.999 + 1.73i)14-s + (0.598 + 1.03i)16-s + 0.325·17-s + 1.44·19-s + (−0.136 − 0.235i)20-s + (−0.434 + 0.752i)22-s + (0.691 − 1.19i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.627665 + 3.55966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627665 + 3.55966i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-25.4 - 44.1i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (3.55e3 - 6.15e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-3.94e4 - 6.83e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-2.06e5 - 3.57e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-1.35e5 + 2.34e5i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 - 1.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.56e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-2.13e7 + 3.69e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-2.58e7 - 4.47e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (6.56e7 - 1.13e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + 4.81e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-4.71e8 + 8.16e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-7.95e8 - 1.37e9i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (1.09e9 + 1.89e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + 3.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (3.22e9 - 5.59e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-9.01e8 - 1.56e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-3.09e9 + 5.35e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 8.00e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.88e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (2.05e10 + 3.55e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (2.66e10 + 4.61e10i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 - 3.21e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (6.37e10 + 1.10e11i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−12.56857595033687094863892799257, −11.60842090483646995367503561132, −10.57082958508168975496229672543, −8.966921706973285240798935338474, −7.73095903227368299057968468993, −6.84412006609771250757522496756, −5.61346095715500995421702383138, −4.74168126944121054764341915438, −3.04609085226774157807012027972, −1.59355837339400598709445637936,
0.836207679508527124495862674272, 1.40227850951797141649352853395, 3.35967164460583086324061555879, 4.19332438091685677042709781874, 5.15553121451319247337863294264, 7.31422946683349201724981583848, 8.084126929932356409427710919544, 9.663785598191097379781757867751, 11.04328363186768767617580359039, 11.48662998471130327187444077668
