| L(s) = 1 | + (−0.980 − 1.16i)5-s + (3.23 − 1.17i)7-s + (3.21 − 3.82i)11-s + (−0.778 − 4.41i)13-s + (3.57 − 2.06i)17-s + (6.75 − 11.7i)19-s + (5.79 − 15.9i)23-s + (3.93 − 22.3i)25-s + (47.1 + 8.30i)29-s + (−14.3 − 5.22i)31-s + (−4.55 − 2.62i)35-s + (−32.3 − 56.0i)37-s + (55.4 − 9.76i)41-s + (22.7 + 19.0i)43-s + (7.04 + 19.3i)47-s + ⋯ |
| L(s) = 1 | + (−0.196 − 0.233i)5-s + (0.462 − 0.168i)7-s + (0.291 − 0.347i)11-s + (−0.0599 − 0.339i)13-s + (0.210 − 0.121i)17-s + (0.355 − 0.615i)19-s + (0.252 − 0.692i)23-s + (0.157 − 0.893i)25-s + (1.62 + 0.286i)29-s + (−0.462 − 0.168i)31-s + (−0.130 − 0.0750i)35-s + (−0.875 − 1.51i)37-s + (1.35 − 0.238i)41-s + (0.528 + 0.443i)43-s + (0.149 + 0.411i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.43529 - 0.743437i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43529 - 0.743437i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.980 + 1.16i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-3.23 + 1.17i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (-3.21 + 3.82i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (0.778 + 4.41i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-3.57 + 2.06i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.75 + 11.7i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-5.79 + 15.9i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-47.1 - 8.30i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 5.22i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (32.3 + 56.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-55.4 + 9.76i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-22.7 - 19.0i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-7.04 - 19.3i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 - 19.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (63.6 + 75.9i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-77.5 + 28.2i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-11.2 - 63.7i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (109. - 63.3i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.0 + 31.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.4 - 82.1i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-16.5 - 2.91i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (66.0 + 38.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-82.1 - 68.9i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−11.18617889163419757445905634750, −10.47960514728550865233485580860, −9.282887133775584307407708567117, −8.428674175421556411088337717238, −7.50172125593341501525021490998, −6.38921004326621193537000441906, −5.16721433619111729690935969240, −4.15502001767362479180038417301, −2.69538286062919894041208256032, −0.856484453072914536322700903461,
1.52916217480547955885354918497, 3.15578648006801536489319132716, 4.44612294580539084184972810375, 5.56794954373565024540326400067, 6.78881278601801519619874160237, 7.70082645889708303238707508675, 8.701964212453583798642443919480, 9.701399037748355905942570704353, 10.63428566974710243417057735905, 11.64982356574039688125208628620
