| L(s) = 1 | + (5.23 + 0.458i)2-s + (−3.90 + 3.43i)3-s + (19.3 + 3.41i)4-s + (5.86 − 9.51i)5-s + (−22.0 + 16.1i)6-s + (9.60 + 6.72i)7-s + (59.1 + 15.8i)8-s + (3.45 − 26.7i)9-s + (35.1 − 47.1i)10-s + (−3.05 + 8.39i)11-s + (−87.1 + 53.0i)12-s + (5.05 + 57.7i)13-s + (47.2 + 39.6i)14-s + (9.74 + 57.2i)15-s + (154. + 56.3i)16-s + (−100. + 27.0i)17-s + ⋯ |
| L(s) = 1 | + (1.85 + 0.162i)2-s + (−0.750 + 0.660i)3-s + (2.41 + 0.426i)4-s + (0.525 − 0.851i)5-s + (−1.49 + 1.10i)6-s + (0.518 + 0.363i)7-s + (2.61 + 0.700i)8-s + (0.127 − 0.991i)9-s + (1.11 − 1.49i)10-s + (−0.0837 + 0.230i)11-s + (−2.09 + 1.27i)12-s + (0.107 + 1.23i)13-s + (0.901 + 0.756i)14-s + (0.167 + 0.985i)15-s + (2.41 + 0.880i)16-s + (−1.44 + 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.01311 + 1.22432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.01311 + 1.22432i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.90 - 3.43i)T \) |
| 5 | \( 1 + (-5.86 + 9.51i)T \) |
| good | 2 | \( 1 + (-5.23 - 0.458i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-9.60 - 6.72i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (3.05 - 8.39i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-5.05 - 57.7i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (100. - 27.0i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-41.3 + 23.8i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (111. + 159. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-78.5 + 65.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-39.8 + 226. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-35.3 - 131. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (212. - 252. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (95.2 + 204. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (45.6 - 65.2i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (51.2 - 51.2i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-57.3 + 20.8i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-128. - 730. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (74.0 - 6.48i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (339. + 196. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (263. - 984. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-436. - 520. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-44.4 + 508. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (409. + 708. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (248. - 115. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.92062802088107754938509558930, −11.86770458266836354380076311302, −11.45308092786716890681214430623, −10.07637250457165357669046842338, −8.619869862172340477055336868043, −6.65947930155852068187821225908, −5.90925193595861164388953552108, −4.67579793592824995985002051536, −4.31410244168352348292538139305, −2.13300392829772178099385351839,
1.83537001769657449194155820138, 3.25978852430480815282641178920, 4.93888062470994223055559137093, 5.81096166232444384072663627132, 6.75105796497017242430175849382, 7.69211361365240197015298214959, 10.35095305367625957819741791294, 11.02382023970335218836373067825, 11.78556242587797942145714180322, 12.85818164293116195435526175071
