| L(s) = 1 | + (−2.99 + 9.20i)2-s + (−21.8 + 15.8i)3-s + (27.6 + 20.1i)4-s + (−60.2 − 185. i)5-s + (−80.7 − 248. i)6-s + (−765. − 556. i)7-s + (−1.27e3 + 923. i)8-s + (225. − 693. i)9-s + 1.88e3·10-s + (3.33e3 − 2.89e3i)11-s − 924.·12-s + (1.59e3 − 4.89e3i)13-s + (7.41e3 − 5.38e3i)14-s + (4.26e3 + 3.09e3i)15-s + (−3.34e3 − 1.02e4i)16-s + (2.41e3 + 7.43e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 + 0.813i)2-s + (−0.467 + 0.339i)3-s + (0.216 + 0.157i)4-s + (−0.215 − 0.663i)5-s + (−0.152 − 0.469i)6-s + (−0.843 − 0.612i)7-s + (−0.877 + 0.637i)8-s + (0.103 − 0.317i)9-s + 0.597·10-s + (0.755 − 0.655i)11-s − 0.154·12-s + (0.200 − 0.617i)13-s + (0.722 − 0.524i)14-s + (0.326 + 0.236i)15-s + (−0.204 − 0.628i)16-s + (0.119 + 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.722379 - 0.309843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.722379 - 0.309843i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (21.8 - 15.8i)T \) |
| 11 | \( 1 + (-3.33e3 + 2.89e3i)T \) |
| good | 2 | \( 1 + (2.99 - 9.20i)T + (-103. - 75.2i)T^{2} \) |
| 5 | \( 1 + (60.2 + 185. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (765. + 556. i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (-1.59e3 + 4.89e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-2.41e3 - 7.43e3i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-4.34e4 + 3.15e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + 1.05e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + (7.43e4 + 5.40e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-1.14e4 + 3.51e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (4.26e5 + 3.09e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (2.76e5 - 2.00e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 - 1.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.39e4 - 1.01e4i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-3.85e5 + 1.18e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.16e5 + 1.57e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (6.56e5 + 2.02e6i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 6.48e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.32e6 - 4.07e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-3.51e6 - 2.55e6i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.57e6 - 4.84e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-4.73e5 - 1.45e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + 9.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.01e6 - 3.13e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−15.61223282246267454027900012308, −13.95552538326962195813814806795, −12.46558471255723810201255249212, −11.33929849440316935759684441872, −9.703082388807906809242078674455, −8.345632420869935254492856262690, −6.88279142829412199859897271756, −5.63481458761590122556744108228, −3.59973288517798851020984726421, −0.41873215248594344055985956073,
1.66678542989091294845408173129, 3.35875876203521533309822411886, 5.99714251594205847790663463175, 7.09988887849488174198729898004, 9.362944841493897310836390707714, 10.36106748285679380403682749055, 11.84974471161431416494716549539, 12.18311966752472071892901991801, 14.05056222330987767534022619166, 15.43650007214791012132315686848
