| L(s) = 1 | + (9.63 − 16.6i)2-s + (8.51 − 14.7i)3-s + (−121. − 210. i)4-s − 381.·5-s + (−164. − 284. i)6-s + (171.5 + 297. i)7-s − 2.22e3·8-s + (948. + 1.64e3i)9-s + (−3.67e3 + 6.36e3i)10-s + (−679. + 1.17e3i)11-s − 4.14e3·12-s + (−5.95e3 + 5.22e3i)13-s + 6.61e3·14-s + (−3.24e3 + 5.62e3i)15-s + (−5.85e3 + 1.01e4i)16-s + (5.04e3 + 8.73e3i)17-s + ⋯ |
| L(s) = 1 | + (0.851 − 1.47i)2-s + (0.182 − 0.315i)3-s + (−0.950 − 1.64i)4-s − 1.36·5-s + (−0.310 − 0.537i)6-s + (0.188 + 0.327i)7-s − 1.53·8-s + (0.433 + 0.751i)9-s + (−1.16 + 2.01i)10-s + (−0.154 + 0.266i)11-s − 0.692·12-s + (−0.751 + 0.659i)13-s + 0.643·14-s + (−0.248 + 0.430i)15-s + (−0.357 + 0.619i)16-s + (0.249 + 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.725473 + 0.166995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.725473 + 0.166995i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (5.95e3 - 5.22e3i)T \) |
| good | 2 | \( 1 + (-9.63 + 16.6i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-8.51 + 14.7i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + 381.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (679. - 1.17e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-5.04e3 - 8.73e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.55e4 + 2.68e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.40e3 - 7.63e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.39e4 - 2.41e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 8.02e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.08e5 - 3.60e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.52e5 - 6.10e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.34e5 - 2.33e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 3.20e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.56e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (7.93e5 + 1.37e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-3.17e5 - 5.49e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.28e5 - 1.60e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.79e5 - 3.11e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.48e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.88e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.61e6 - 6.26e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (4.84e6 + 8.39e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.57100540765082772248036676235, −11.77856290870587160474828318851, −11.03865293631932464500973517061, −9.900220013569572258851117781933, −8.350367462647222795826717927904, −7.13139611471260768309403273265, −4.97529347881885590121556640749, −4.18717555182992971416008071341, −2.80693912668267824247029727416, −1.58512214579299654654257141291,
0.17474012833738527182007240299, 3.52454721438222140832156695178, 4.23462467353077844626059688381, 5.50465046176580265873269288514, 7.02728257735529801941072496666, 7.70274971403861362082523583283, 8.722939597604737344625229647758, 10.40919288196405603106494609029, 12.01612896671528693576950876391, 12.70571464368377182458248062043
