| L(s) = 1 | + (−7.14 + 4.59i)3-s + (−4.30 − 9.42i)5-s + (54.2 − 15.9i)7-s + (−70.9 + 155. i)9-s + (−60.7 − 70.1i)11-s + (−336. − 98.9i)13-s + (73.9 + 47.5i)15-s + (297. − 2.06e3i)17-s + (−62.3 − 433. i)19-s + (−314. + 363. i)21-s + (−30.3 − 2.53e3i)23-s + (1.97e3 − 2.28e3i)25-s + (−500. − 3.47e3i)27-s + (−294. + 2.04e3i)29-s + (−6.24e3 − 4.01e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.458 + 0.294i)3-s + (−0.0769 − 0.168i)5-s + (0.418 − 0.122i)7-s + (−0.292 + 0.639i)9-s + (−0.151 − 0.174i)11-s + (−0.552 − 0.162i)13-s + (0.0848 + 0.0545i)15-s + (0.249 − 1.73i)17-s + (−0.0396 − 0.275i)19-s + (−0.155 + 0.179i)21-s + (−0.0119 − 0.999i)23-s + (0.632 − 0.729i)25-s + (−0.132 − 0.918i)27-s + (−0.0649 + 0.451i)29-s + (−1.16 − 0.750i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.588581 - 0.679131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.588581 - 0.679131i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (30.3 + 2.53e3i)T \) |
| good | 3 | \( 1 + (7.14 - 4.59i)T + (100. - 221. i)T^{2} \) |
| 5 | \( 1 + (4.30 + 9.42i)T + (-2.04e3 + 2.36e3i)T^{2} \) |
| 7 | \( 1 + (-54.2 + 15.9i)T + (1.41e4 - 9.08e3i)T^{2} \) |
| 11 | \( 1 + (60.7 + 70.1i)T + (-2.29e4 + 1.59e5i)T^{2} \) |
| 13 | \( 1 + (336. + 98.9i)T + (3.12e5 + 2.00e5i)T^{2} \) |
| 17 | \( 1 + (-297. + 2.06e3i)T + (-1.36e6 - 4.00e5i)T^{2} \) |
| 19 | \( 1 + (62.3 + 433. i)T + (-2.37e6 + 6.97e5i)T^{2} \) |
| 29 | \( 1 + (294. - 2.04e3i)T + (-1.96e7 - 5.77e6i)T^{2} \) |
| 31 | \( 1 + (6.24e3 + 4.01e3i)T + (1.18e7 + 2.60e7i)T^{2} \) |
| 37 | \( 1 + (-4.19e3 + 9.17e3i)T + (-4.54e7 - 5.24e7i)T^{2} \) |
| 41 | \( 1 + (-316. - 694. i)T + (-7.58e7 + 8.75e7i)T^{2} \) |
| 43 | \( 1 + (7.74e3 - 4.97e3i)T + (6.10e7 - 1.33e8i)T^{2} \) |
| 47 | \( 1 + 9.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.85e4 + 8.38e3i)T + (3.51e8 - 2.26e8i)T^{2} \) |
| 59 | \( 1 + (7.86e3 + 2.30e3i)T + (6.01e8 + 3.86e8i)T^{2} \) |
| 61 | \( 1 + (-5.63e3 - 3.62e3i)T + (3.50e8 + 7.68e8i)T^{2} \) |
| 67 | \( 1 + (-930. + 1.07e3i)T + (-1.92e8 - 1.33e9i)T^{2} \) |
| 71 | \( 1 + (1.44e4 - 1.66e4i)T + (-2.56e8 - 1.78e9i)T^{2} \) |
| 73 | \( 1 + (-6.81e3 - 4.74e4i)T + (-1.98e9 + 5.84e8i)T^{2} \) |
| 79 | \( 1 + (-2.75e4 - 8.09e3i)T + (2.58e9 + 1.66e9i)T^{2} \) |
| 83 | \( 1 + (3.80e4 - 8.33e4i)T + (-2.57e9 - 2.97e9i)T^{2} \) |
| 89 | \( 1 + (-6.39e4 + 4.11e4i)T + (2.31e9 - 5.07e9i)T^{2} \) |
| 97 | \( 1 + (4.13e3 + 9.05e3i)T + (-5.62e9 + 6.48e9i)T^{2} \) |
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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
−12.73927390238194171342992077400, −11.56938416604228011902257038797, −10.79658813816145235325298166872, −9.612766550735735208353370441627, −8.286510721934009854975625046423, −7.12406485815913856946958122937, −5.46042895707427541036345511284, −4.58117632003368084224203055969, −2.56963910741586884545490264652, −0.38642455630537674345015378047,
1.56520459256704456833175817735, 3.55134601120161229754577687887, 5.26450011311497926330441046561, 6.43072198561875065390956260194, 7.67547147761005193974373667047, 8.938151933757543249533337833208, 10.26129001058863420401737230093, 11.38294134970274625644757268489, 12.24987492607035480579598607204, 13.19968321021743691608912095768
