| L(s) = 1 | + (−10.1 + 17.5i)2-s + (−14.5 + 25.1i)3-s + (−141. − 245. i)4-s + 326.·5-s + (−295. − 511. i)6-s + (171.5 + 297. i)7-s + 3.15e3·8-s + (670. + 1.16e3i)9-s + (−3.30e3 + 5.73e3i)10-s + (−1.36e3 + 2.37e3i)11-s + 8.24e3·12-s + (−5.68e3 − 5.51e3i)13-s − 6.95e3·14-s + (−4.74e3 + 8.22e3i)15-s + (−1.38e4 + 2.39e4i)16-s + (6.98e3 + 1.20e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.896 + 1.55i)2-s + (−0.311 + 0.538i)3-s + (−1.10 − 1.91i)4-s + 1.16·5-s + (−0.557 − 0.965i)6-s + (0.188 + 0.327i)7-s + 2.17·8-s + (0.306 + 0.530i)9-s + (−1.04 + 1.81i)10-s + (−0.310 + 0.537i)11-s + 1.37·12-s + (−0.717 − 0.696i)13-s − 0.677·14-s + (−0.363 + 0.628i)15-s + (−0.845 + 1.46i)16-s + (0.344 + 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.178071 - 0.101749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178071 - 0.101749i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 - 297. i)T \) |
| 13 | \( 1 + (5.68e3 + 5.51e3i)T \) |
| good | 2 | \( 1 + (10.1 - 17.5i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (14.5 - 25.1i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 - 326.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.36e3 - 2.37e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-6.98e3 - 1.20e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.83e4 + 4.91e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.10e4 - 3.65e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.60e4 - 7.97e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.59e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.26e5 - 2.18e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.61e5 - 2.79e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.61e5 + 4.53e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 - 6.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.37e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.09e6 + 1.89e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.19e6 + 2.06e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.04e6 + 3.54e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.83e6 - 3.17e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.70e6 - 6.41e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-2.35e6 - 4.08e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.90112698111894615924892799325, −12.86628846251662950155969673764, −10.74671870196151874418599199592, −9.961714375608277960755510380640, −9.186324673917507751421728378245, −7.927967051171594990626286466827, −6.80085713434091304856582456296, −5.53265617276333595170062198062, −4.94366617910561034013596665898, −1.91656524769560894755524449961,
0.090934489924799555789684314826, 1.42069226951250127613745981479, 2.28422959507640283751420002868, 3.95732779005034764196872344979, 5.92652122065459206630017599414, 7.48036561891624441096128145939, 8.869740437652613340719758307556, 9.850780189759279546886146554480, 10.51353378723115131394500429081, 11.78051379853695116416490497689
