| L(s) = 1 | + (12.8 + 4.18i)2-s + (12.6 − 9.16i)3-s + (96.7 + 70.2i)4-s + (19.1 + 58.8i)5-s + (200. − 65.2i)6-s + (−53.6 + 73.8i)7-s + (442. + 608. i)8-s + (75.0 − 231. i)9-s + 837. i·10-s + (743. − 1.10e3i)11-s + 1.86e3·12-s + (−6.65 − 2.16i)13-s + (−1.00e3 + 727. i)14-s + (779. + 566. i)15-s + (785. + 2.41e3i)16-s + (−8.32e3 + 2.70e3i)17-s + ⋯ |
| L(s) = 1 | + (1.61 + 0.523i)2-s + (0.467 − 0.339i)3-s + (1.51 + 1.09i)4-s + (0.152 + 0.470i)5-s + (0.929 − 0.302i)6-s + (−0.156 + 0.215i)7-s + (0.863 + 1.18i)8-s + (0.103 − 0.317i)9-s + 0.837i·10-s + (0.558 − 0.829i)11-s + 1.07·12-s + (−0.00302 − 0.000983i)13-s + (−0.364 + 0.264i)14-s + (0.231 + 0.167i)15-s + (0.191 + 0.590i)16-s + (−1.69 + 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.10207 + 1.36295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.10207 + 1.36295i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-12.6 + 9.16i)T \) |
| 11 | \( 1 + (-743. + 1.10e3i)T \) |
| good | 2 | \( 1 + (-12.8 - 4.18i)T + (51.7 + 37.6i)T^{2} \) |
| 5 | \( 1 + (-19.1 - 58.8i)T + (-1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (53.6 - 73.8i)T + (-3.63e4 - 1.11e5i)T^{2} \) |
| 13 | \( 1 + (6.65 + 2.16i)T + (3.90e6 + 2.83e6i)T^{2} \) |
| 17 | \( 1 + (8.32e3 - 2.70e3i)T + (1.95e7 - 1.41e7i)T^{2} \) |
| 19 | \( 1 + (1.48e3 + 2.04e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + 5.00e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-3.88e3 + 5.34e3i)T + (-1.83e8 - 5.65e8i)T^{2} \) |
| 31 | \( 1 + (4.08e3 - 1.25e4i)T + (-7.18e8 - 5.21e8i)T^{2} \) |
| 37 | \( 1 + (7.11e4 + 5.16e4i)T + (7.92e8 + 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-5.37e4 - 7.39e4i)T + (-1.46e9 + 4.51e9i)T^{2} \) |
| 43 | \( 1 - 5.99e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.43e5 + 1.04e5i)T + (3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (5.74e4 - 1.76e5i)T + (-1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-2.50e5 - 1.81e5i)T + (1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 + (-8.61e4 + 2.79e4i)T + (4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 + 2.62e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-1.20e5 - 3.70e5i)T + (-1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (1.53e5 - 2.11e5i)T + (-4.67e10 - 1.43e11i)T^{2} \) |
| 79 | \( 1 + (-2.44e3 - 795. i)T + (1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-3.69e5 + 1.20e5i)T + (2.64e11 - 1.92e11i)T^{2} \) |
| 89 | \( 1 - 8.17e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (4.48e4 - 1.37e5i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
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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
−15.18582508053550963863115109891, −14.22203881865935493753435539228, −13.42744601042981042377808949267, −12.35127810090450197096852335993, −11.00224656774247633283448934444, −8.762215022748871487419024925268, −6.96473344176603004739935962310, −6.02354643867004061407177077743, −4.12483164254579006027817460044, −2.62625859910320494613285006768,
2.11763703139565553723174315803, 3.89830231756023725499384643800, 4.99564538082100558911860019932, 6.77978758805048251935481729030, 9.042720999090636013728822735861, 10.59075889451959710225809161423, 11.94962979807742524219632820940, 13.03219374604920242176116811320, 13.91752273766760482425483317456, 14.99041420304152775201849255636
