| 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
−9.631721988827806463573861451805, −8.998478489776215270551079493122, −8.262452078136398123859553715080, −7.25677391989420332418454298333, −6.67880321499723611867918048093, −5.53216496866485295078121762577, −4.33471299569617165239839605368, −3.46390557702401994933592155441, −2.27950946036695778358939070885, −0.67693479891366614849510658656,
1.68541635117576189282188468071, 3.04734001692085481893466644626, 4.05665529030622854078750256552, 4.71849654159004798679600694783, 6.25264347020485958748810959122, 6.75831569210778706142141549556, 8.119980631711133391743597785575, 8.472752187288544711155612579597, 9.517644988719473721495445463562, 10.28032596429775720484148372004
