| L(s) = 1 | + (−10.4 − 4.31i)2-s + (41.7 + 21.0i)3-s + (90.8 + 90.1i)4-s + (−194. − 336. i)5-s + (−346. − 400. i)6-s + (−965. − 557. i)7-s + (−561. − 1.33e3i)8-s + (1.30e3 + 1.75e3i)9-s + (580. + 4.35e3i)10-s + (3.10e3 + 1.79e3i)11-s + (1.89e3 + 5.67e3i)12-s + (−2.08e3 + 1.20e3i)13-s + (7.69e3 + 9.99e3i)14-s + (−1.03e3 − 1.81e4i)15-s + (112. + 1.63e4i)16-s − 1.02e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.924 − 0.381i)2-s + (0.893 + 0.449i)3-s + (0.709 + 0.704i)4-s + (−0.694 − 1.20i)5-s + (−0.654 − 0.756i)6-s + (−1.06 − 0.614i)7-s + (−0.387 − 0.921i)8-s + (0.595 + 0.803i)9-s + (0.183 + 1.37i)10-s + (0.702 + 0.405i)11-s + (0.316 + 0.948i)12-s + (−0.263 + 0.152i)13-s + (0.749 + 0.973i)14-s + (−0.0794 − 1.38i)15-s + (0.00689 + 0.999i)16-s − 0.0507i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.257851 + 0.363913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257851 + 0.363913i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.4 + 4.31i)T \) |
| 3 | \( 1 + (-41.7 - 21.0i)T \) |
| good | 5 | \( 1 + (194. + 336. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (965. + 557. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.10e3 - 1.79e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (2.08e3 - 1.20e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.02e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.98e4 - 6.90e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.96e4 - 5.14e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.08e5 - 6.27e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.61e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (-2.10e5 + 1.21e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.54e5 - 7.86e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (5.26e5 - 9.11e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.96e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (5.77e4 - 3.33e4i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.70e6 - 1.55e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.64e5 + 1.15e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.77e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.86e6 + 2.23e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.84e6 - 3.37e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 2.74e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (3.42e6 - 5.93e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.12801600803636533959431570878, −12.53417254718050860773718025169, −11.09674795806538839214827962263, −9.707655834676986393559502641922, −9.154613130307882279358024197273, −8.062878962745752709362004872492, −6.94183886806817231232477123719, −4.37358468823352904022242313720, −3.32346338021136627032186972497, −1.42671640874984362248137413844,
0.18720706469182618421434420588, 2.32281755838058792954706775258, 3.42670327725934143760733593288, 6.37676823554319087972636087387, 6.94538157219217443649993495627, 8.240369241661975666745718733226, 9.206081770900183722210925065884, 10.34695701697836718478191564207, 11.58668999012971071340292079695, 12.83724698552727772487789803315
