| L(s) = 1 | + (0.766 + 0.642i)3-s + (−1.03 + 0.375i)5-s + (−0.450 + 0.260i)7-s + (0.173 + 0.984i)9-s + (2.93 + 1.69i)11-s + (−0.0473 − 0.0564i)13-s + (−1.03 − 0.375i)15-s + (−1.06 + 6.06i)17-s + (−3.88 + 1.98i)19-s + (−0.512 − 0.0903i)21-s + (2.21 − 6.07i)23-s + (−2.90 + 2.43i)25-s + (−0.500 + 0.866i)27-s + (5.19 − 0.916i)29-s + (1.91 + 3.31i)31-s + ⋯ |
| L(s) = 1 | + (0.442 + 0.371i)3-s + (−0.461 + 0.167i)5-s + (−0.170 + 0.0983i)7-s + (0.0578 + 0.328i)9-s + (0.886 + 0.511i)11-s + (−0.0131 − 0.0156i)13-s + (−0.266 − 0.0969i)15-s + (−0.259 + 1.46i)17-s + (−0.890 + 0.454i)19-s + (−0.111 − 0.0197i)21-s + (0.460 − 1.26i)23-s + (−0.581 + 0.487i)25-s + (−0.0962 + 0.166i)27-s + (0.965 − 0.170i)29-s + (0.343 + 0.595i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0998 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0998 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.985520 + 1.08933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.985520 + 1.08933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (3.88 - 1.98i)T \) |
| good | 5 | \( 1 + (1.03 - 0.375i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.450 - 0.260i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0473 + 0.0564i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.06 - 6.06i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.21 + 6.07i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-5.19 + 0.916i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.91 - 3.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.69iT - 37T^{2} \) |
| 41 | \( 1 + (4.91 - 5.85i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.06 - 2.91i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.25 + 0.220i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.236 - 0.651i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.67 + 9.51i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.35 - 1.21i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.551 + 3.12i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.327 + 0.119i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.94 - 5.83i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (9.86 + 8.28i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 6.95i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.02 - 2.41i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (4.75 + 0.838i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.28032596429775720484148372004, −9.517644988719473721495445463562, −8.472752187288544711155612579597, −8.119980631711133391743597785575, −6.75831569210778706142141549556, −6.25264347020485958748810959122, −4.71849654159004798679600694783, −4.05665529030622854078750256552, −3.04734001692085481893466644626, −1.68541635117576189282188468071,
0.67693479891366614849510658656, 2.27950946036695778358939070885, 3.46390557702401994933592155441, 4.33471299569617165239839605368, 5.53216496866485295078121762577, 6.67880321499723611867918048093, 7.25677391989420332418454298333, 8.262452078136398123859553715080, 8.998478489776215270551079493122, 9.631721988827806463573861451805
