| L(s) = 1 | + (−2.88 − 0.812i)3-s + (0.980 + 1.16i)5-s + (−3.23 + 1.17i)7-s + (7.67 + 4.69i)9-s + (3.21 − 3.82i)11-s + (−0.778 − 4.41i)13-s + (−1.88 − 4.17i)15-s + (−3.57 + 2.06i)17-s + (−6.75 + 11.7i)19-s + (10.3 − 0.772i)21-s + (5.79 − 15.9i)23-s + (3.93 − 22.3i)25-s + (−18.3 − 19.7i)27-s + (−47.1 − 8.30i)29-s + (14.3 + 5.22i)31-s + ⋯ |
| L(s) = 1 | + (−0.962 − 0.270i)3-s + (0.196 + 0.233i)5-s + (−0.462 + 0.168i)7-s + (0.853 + 0.521i)9-s + (0.291 − 0.347i)11-s + (−0.0599 − 0.339i)13-s + (−0.125 − 0.278i)15-s + (−0.210 + 0.121i)17-s + (−0.355 + 0.615i)19-s + (0.491 − 0.0367i)21-s + (0.252 − 0.692i)23-s + (0.157 − 0.893i)25-s + (−0.680 − 0.733i)27-s + (−1.62 − 0.286i)29-s + (0.462 + 0.168i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.154534 - 0.430194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.154534 - 0.430194i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.88 + 0.812i)T \) |
| 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
−10.62728536586074896211296554402, −9.943956426002443991763516238438, −8.795356833518774439220273803437, −7.66515449471019497410438693647, −6.58454212939525496912698645449, −6.00278431578741597101704108167, −4.92313041554689468863383620126, −3.60277666757519593770932823415, −1.97542562801495988530267306563, −0.21906511523938113592975509276,
1.53159162435285573653329008842, 3.47239617335918124331707517778, 4.64870364384607527138280495225, 5.51872366827909100109705165589, 6.63518362832004492869921006342, 7.26445022224196131411858677152, 8.836076834146300136666991686666, 9.612061246332474537430862784587, 10.37406340119457656052132779651, 11.39627081273770164310203156122
