| L(s) = 1 | + (−6.09 − 10.5i)2-s + (−33.3 + 32.8i)3-s + (−10.3 + 17.9i)4-s + (−246. + 426. i)5-s + (549. + 151. i)6-s + (−382. − 662. i)7-s − 1.30e3·8-s + (32.3 − 2.18e3i)9-s + 6.00e3·10-s + (36.3 + 62.9i)11-s + (−243. − 937. i)12-s + (−3.01e3 + 5.21e3i)13-s + (−4.66e3 + 8.07e3i)14-s + (−5.79e3 − 2.22e4i)15-s + (9.30e3 + 1.61e4i)16-s − 5.98e3·17-s + ⋯ |
| L(s) = 1 | + (−0.538 − 0.933i)2-s + (−0.712 + 0.701i)3-s + (−0.0808 + 0.140i)4-s + (−0.880 + 1.52i)5-s + (1.03 + 0.286i)6-s + (−0.421 − 0.729i)7-s − 0.903·8-s + (0.0147 − 0.999i)9-s + 1.89·10-s + (0.00823 + 0.0142i)11-s + (−0.0407 − 0.156i)12-s + (−0.380 + 0.658i)13-s + (−0.454 + 0.786i)14-s + (−0.443 − 1.70i)15-s + (0.567 + 0.983i)16-s − 0.295·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0753785 + 0.158576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0753785 + 0.158576i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (33.3 - 32.8i)T \) |
| good | 2 | \( 1 + (6.09 + 10.5i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (246. - 426. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (382. + 662. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-36.3 - 62.9i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.01e3 - 5.21e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 5.98e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (1.21e4 - 2.10e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.33e4 + 7.51e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.05e5 - 1.83e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.96e5 - 3.39e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.43e5 - 5.95e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.20e5 + 5.55e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 8.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.25e6 + 2.18e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.21e5 - 3.83e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.96e5 - 5.12e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.48e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-4.44e5 - 7.70e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.69e6 - 2.93e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 1.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.30e6 - 7.44e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−20.06111842910270710758850642414, −19.01437211258988668084454783429, −17.79603401294061415906736936644, −15.94171808353352776451410361598, −14.64330036050327560860198215390, −11.79557141929586599310214785561, −10.89109953509779145011824351881, −9.842218563520984591037489283559, −6.78167854361257799659840712557, −3.48922696596644972508821272999,
0.17278031352284216741456784971, 5.52567797733740367884589999690, 7.54489391620336866434762905576, 8.846049821215955760026441489515, 11.93373301046332620978097072101, 12.75099448236535670252621905583, 15.58700305287893886714316408437, 16.43208138802770944500171832368, 17.49984738062113145356072843250, 18.93758677929228791390340866019
