| L(s) = 1 | + (−0.162 + 1.85i)2-s + (0.207 − 1.71i)3-s + (−1.46 − 0.257i)4-s + (1.63 − 1.52i)5-s + (3.16 + 0.665i)6-s + (1.79 − 2.56i)7-s + (−0.248 + 0.928i)8-s + (−2.91 − 0.712i)9-s + (2.57 + 3.28i)10-s + (−1.21 + 3.34i)11-s + (−0.746 + 2.46i)12-s + (1.89 − 0.165i)13-s + (4.47 + 3.75i)14-s + (−2.29 − 3.12i)15-s + (−4.47 − 1.62i)16-s + (0.886 + 3.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.115 + 1.31i)2-s + (0.119 − 0.992i)3-s + (−0.731 − 0.128i)4-s + (0.729 − 0.684i)5-s + (1.29 + 0.271i)6-s + (0.678 − 0.969i)7-s + (−0.0879 + 0.328i)8-s + (−0.971 − 0.237i)9-s + (0.815 + 1.03i)10-s + (−0.367 + 1.00i)11-s + (−0.215 + 0.710i)12-s + (0.525 − 0.0460i)13-s + (1.19 + 1.00i)14-s + (−0.591 − 0.806i)15-s + (−1.11 − 0.407i)16-s + (0.215 + 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16056 + 0.285010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16056 + 0.285010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.207 + 1.71i)T \) |
| 5 | \( 1 + (-1.63 + 1.52i)T \) |
| good | 2 | \( 1 + (0.162 - 1.85i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.79 + 2.56i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.21 - 3.34i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 0.165i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.886 - 3.30i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.00 - 2.89i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.06 + 0.749i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (0.0144 - 0.0121i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.260 + 1.47i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-9.77 + 2.61i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (6.39 - 7.61i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.93 - 3.23i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (9.02 + 6.31i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (3.54 + 3.54i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.27 + 1.91i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.47 - 8.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.0194 - 0.222i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (0.428 + 0.247i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.75 - 2.07i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.13 + 1.35i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (7.35 + 0.643i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (4.43 + 7.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.23 - 6.93i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−13.36660318113245403158570217115, −12.79600160927848732264846055198, −11.37282105323431670806298382934, −9.997148028887738122553934376151, −8.451813152277653641937624902341, −7.981582489887913868206844160688, −6.82074581681488545970501681074, −5.93655814421557101187407804288, −4.64642748512433910088882085030, −1.80019438354750366746866136480,
2.34098343657538675132788786558, 3.31784309726278285331064957780, 5.03201166338849681133331170127, 6.28567884323584934913059897510, 8.476941575712997353613088088370, 9.295294061374876147907165430022, 10.29534524029057782308794582761, 11.14534496217615476788216379528, 11.56101059757629536764166667131, 13.09338756957537554070624390038
