| L(s) = 1 | + (−18.2 − 13.2i)2-s + (−8.34 + 25.6i)3-s + (117. + 362. i)4-s + (−299. + 217. i)5-s + (493. − 358. i)6-s + (233. + 720. i)7-s + (1.76e3 − 5.44e3i)8-s + (−589. − 428. i)9-s + 8.34e3·10-s + (−2.65e3 + 3.52e3i)11-s − 1.03e4·12-s + (1.64e3 + 1.19e3i)13-s + (5.28e3 − 1.62e4i)14-s + (−3.08e3 − 9.49e3i)15-s + (−6.49e4 + 4.72e4i)16-s + (−3.80e3 + 2.76e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 − 1.17i)2-s + (−0.178 + 0.549i)3-s + (0.921 + 2.83i)4-s + (−1.07 + 0.777i)5-s + (0.931 − 0.677i)6-s + (0.257 + 0.793i)7-s + (1.22 − 3.75i)8-s + (−0.269 − 0.195i)9-s + 2.63·10-s + (−0.601 + 0.799i)11-s − 1.72·12-s + (0.207 + 0.151i)13-s + (0.514 − 1.58i)14-s + (−0.235 − 0.726i)15-s + (−3.96 + 2.88i)16-s + (−0.187 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00825513 - 0.0509749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00825513 - 0.0509749i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 - 25.6i)T \) |
| 11 | \( 1 + (2.65e3 - 3.52e3i)T \) |
| good | 2 | \( 1 + (18.2 + 13.2i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (299. - 217. i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-233. - 720. i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-1.64e3 - 1.19e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (3.80e3 - 2.76e3i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (150. - 461. i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (3.73e4 + 1.14e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (8.36e4 + 6.07e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.37e5 + 4.24e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-4.14e4 + 1.27e5i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + 4.12e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.12e5 - 6.55e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-9.27e5 - 6.74e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (3.66e5 + 1.12e6i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.03e6 + 1.47e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 - 2.82e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (2.37e6 - 1.72e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.13e6 - 3.47e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (8.69e4 + 6.31e4i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-4.17e5 + 3.03e5i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 + 3.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (5.04e6 + 3.66e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−15.99062687889858124653468747829, −15.24525076207771775618152276579, −12.61100912383213125916806725969, −11.56394866071735051729374862089, −10.86080443710752096686081923156, −9.660920675766667929912863596243, −8.374631463049243182549766458215, −7.23878775963502911618423903903, −3.84964169704573464108130607977, −2.36611499120029162128437604829,
0.04939001019610945437774045196, 1.10776427531540377793922022575, 5.21343875009284475350494072929, 6.91384677794359275361232519484, 7.966342598185701216767739671450, 8.692621538307609626772492958117, 10.48438936944230953186221702543, 11.52497759912361996829201767318, 13.59935302477179277785197779173, 15.05245190155679053249551697113
