| L(s) = 1 | + (0.766 − 0.642i)2-s + (1.25 − 1.19i)3-s + (0.173 − 0.984i)4-s + (0.298 − 0.819i)5-s + (0.197 − 1.72i)6-s + (−0.0257 − 0.0446i)7-s + (−0.500 − 0.866i)8-s + (0.161 − 2.99i)9-s + (−0.298 − 0.819i)10-s + 2.87i·11-s + (−0.954 − 1.44i)12-s + (0.143 + 0.393i)13-s + (−0.0484 − 0.0176i)14-s + (−0.600 − 1.38i)15-s + (−0.939 − 0.342i)16-s + (−1.02 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.725 − 0.687i)3-s + (0.0868 − 0.492i)4-s + (0.133 − 0.366i)5-s + (0.0805 − 0.702i)6-s + (−0.00973 − 0.0168i)7-s + (−0.176 − 0.306i)8-s + (0.0537 − 0.998i)9-s + (−0.0942 − 0.258i)10-s + 0.867i·11-s + (−0.275 − 0.417i)12-s + (0.0396 + 0.109i)13-s + (−0.0129 − 0.00470i)14-s + (−0.155 − 0.357i)15-s + (−0.234 − 0.0855i)16-s + (−0.248 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55904 - 1.52556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55904 - 1.52556i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-1.25 + 1.19i)T \) |
| 19 | \( 1 + (2.25 + 3.72i)T \) |
| good | 5 | \( 1 + (-0.298 + 0.819i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0257 + 0.0446i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.87iT - 11T^{2} \) |
| 13 | \( 1 + (-0.143 - 0.393i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.02 - 2.81i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.73 - 0.657i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.806 - 4.57i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 3.21iT - 31T^{2} \) |
| 37 | \( 1 + 9.55iT - 37T^{2} \) |
| 41 | \( 1 + (2.97 - 2.49i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.38 - 7.86i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.24 + 0.395i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.01 - 1.69i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.82 - 10.3i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.22 + 1.53i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.719 + 0.857i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.26 - 2.74i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.07 + 6.10i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.461 + 1.26i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (10.0 - 5.82i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0172 + 0.0976i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.00 - 10.7i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.46530160746722201321933318004, −10.46959246626847573299917564717, −9.287221446874993951059281319341, −8.693224789005668674307974873140, −7.33429587942232136223668776584, −6.58212173435231198440503242971, −5.19605471275659576749429716782, −4.03042729101264629806392743955, −2.71507478308840246481452590559, −1.48349165235510757103696971727,
2.55897720152318099185203160648, 3.57823702344532646111615237106, 4.69612200659134773085809657372, 5.81162435985271123147940576027, 6.93291244787825973247573857257, 8.097564969484578458556610235101, 8.774030339637818591382348515516, 9.901884942879244359681265929657, 10.78747690307000329510144780433, 11.69400421736018595736694358679
