| L(s) = 1 | + (−10.7 + 18.5i)2-s + (21.6 + 41.4i)3-s + (−165. − 286. i)4-s + (32.6 + 56.5i)5-s + (−1.00e3 − 42.3i)6-s + (−118. + 204. i)7-s + 4.33e3·8-s + (−1.24e3 + 1.79e3i)9-s − 1.39e3·10-s + (−2.30e3 + 3.99e3i)11-s + (8.28e3 − 1.30e4i)12-s + (68.1 + 118. i)13-s + (−2.53e3 − 4.38e3i)14-s + (−1.63e3 + 2.57e3i)15-s + (−2.52e4 + 4.37e4i)16-s + 5.31e3·17-s + ⋯ |
| L(s) = 1 | + (−0.946 + 1.63i)2-s + (0.462 + 0.886i)3-s + (−1.29 − 2.23i)4-s + (0.116 + 0.202i)5-s + (−1.89 − 0.0800i)6-s + (−0.130 + 0.225i)7-s + 2.99·8-s + (−0.571 + 0.820i)9-s − 0.441·10-s + (−0.522 + 0.904i)11-s + (1.38 − 2.17i)12-s + (0.00860 + 0.0148i)13-s + (−0.246 − 0.426i)14-s + (−0.125 + 0.197i)15-s + (−1.54 + 2.66i)16-s + 0.262·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.107705 - 0.809740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.107705 - 0.809740i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-21.6 - 41.4i)T \) |
| good | 2 | \( 1 + (10.7 - 18.5i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-32.6 - 56.5i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (118. - 204. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.30e3 - 3.99e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-68.1 - 118. i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 5.31e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.64e4 - 6.31e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.56e4 - 6.18e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-2.88e4 - 4.99e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 8.84e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.27e5 + 2.21e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.24e5 + 5.62e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-5.22e5 + 9.05e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 7.77e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.01e6 + 1.76e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (9.27e5 - 1.60e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.81e6 - 3.14e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.93e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-3.13e6 + 5.42e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.68e5 + 9.85e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 4.28e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.85e6 + 1.01e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−20.21871452155542625768468199081, −18.70934930484069957752053940033, −17.41371508186713224341538831574, −16.04169026272093013107113995630, −15.20292052259159289403554014641, −13.97512943732577657314397028635, −10.25499083356459832008044179874, −9.110813385296570424771038872299, −7.47532045466587310908824966677, −5.26670876879100864170825092192,
0.908209965280192190178791306045, 2.98222066233880410369162635926, 7.908225702887590896482523542961, 9.324736888658053328156560982380, 11.13854277624499975506579568947, 12.59273156079223298618013725319, 13.68860843910493886486591298692, 16.83878312694164878671736236240, 18.26026700454729875682164873026, 19.02777238140979652179753839813
