| L(s) = 1 | + (−3.66 + 2.11i)2-s + (31.3 + 54.3i)3-s + (−55.0 + 95.3i)4-s + 422. i·5-s + (−229. − 132. i)6-s + (−297. − 171.5i)7-s − 1.00e3i·8-s + (−873. + 1.51e3i)9-s + (−894. − 1.54e3i)10-s + (−5.39e3 + 3.11e3i)11-s − 6.90e3·12-s + (−2.59e3 + 7.48e3i)13-s + 1.45e3·14-s + (−2.29e4 + 1.32e4i)15-s + (−4.91e3 − 8.51e3i)16-s + (1.47e4 − 2.55e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.323 + 0.187i)2-s + (0.670 + 1.16i)3-s + (−0.430 + 0.744i)4-s + 1.51i·5-s + (−0.434 − 0.250i)6-s + (−0.327 − 0.188i)7-s − 0.695i·8-s + (−0.399 + 0.692i)9-s + (−0.282 − 0.489i)10-s + (−1.22 + 0.705i)11-s − 1.15·12-s + (−0.327 + 0.944i)13-s + 0.141·14-s + (−1.75 + 1.01i)15-s + (−0.299 − 0.519i)16-s + (0.729 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.710716 - 1.01273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.710716 - 1.01273i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (297. + 171.5i)T \) |
| 13 | \( 1 + (2.59e3 - 7.48e3i)T \) |
| good | 2 | \( 1 + (3.66 - 2.11i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-31.3 - 54.3i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 422. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + (5.39e3 - 3.11e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.47e4 + 2.55e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.06e4 - 1.76e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.89e4 - 6.74e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.89e4 - 5.01e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 9.53e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (2.06e3 - 1.19e3i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.15e5 + 1.24e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.06e5 + 5.30e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 7.87e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.74e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.45e6 + 1.41e6i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-6.99e5 + 1.21e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.35e5 + 7.82e4i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-4.31e6 - 2.49e6i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 1.27e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.02e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.17e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.13e6 + 1.23e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-9.16e6 - 5.28e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.86283840991270825669804955895, −12.31655078021517111606817376880, −10.93016617657600583436285481705, −9.777215282242881763255818773209, −9.442646342974535456236330396190, −7.70729755776688309640397238783, −7.07339067718309321156855718319, −4.90811786283425882725612270001, −3.47034506008524393466023925314, −2.85576516252595876772497875106,
0.45717934651536070792136690079, 1.20040751118547088203290792452, 2.69441449288306268470385594872, 4.95155628849191144435713765025, 5.91701891188671322318219597878, 7.913780132467481704433531393591, 8.394916735080548597288820247153, 9.474853668297227054290162915017, 10.68876533605901223888105521730, 12.41402718786440574080732087223
