| L(s) = 1 | + (−1.73 − 0.0606i)3-s + (−0.00848 − 0.00308i)5-s + (0.356 − 2.02i)7-s + (2.99 + 0.210i)9-s + (−3.47 + 1.26i)11-s + (−1.25 + 1.05i)13-s + (0.0144 + 0.00586i)15-s + (−3.38 − 5.86i)17-s + (1.25 − 2.16i)19-s + (−0.739 + 3.47i)21-s + (−0.120 − 0.680i)23-s + (−3.83 − 3.21i)25-s + (−5.16 − 0.545i)27-s + (−2.53 − 2.13i)29-s + (−1.38 − 7.83i)31-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0350i)3-s + (−0.00379 − 0.00138i)5-s + (0.134 − 0.763i)7-s + (0.997 + 0.0700i)9-s + (−1.04 + 0.381i)11-s + (−0.348 + 0.292i)13-s + (0.00374 + 0.00151i)15-s + (−0.821 − 1.42i)17-s + (0.287 − 0.497i)19-s + (−0.161 + 0.758i)21-s + (−0.0250 − 0.141i)23-s + (−0.766 − 0.642i)25-s + (−0.994 − 0.104i)27-s + (−0.471 − 0.395i)29-s + (−0.248 − 1.40i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.219431 - 0.463768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.219431 - 0.463768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0606i)T \) |
| good | 5 | \( 1 + (0.00848 + 0.00308i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.356 + 2.02i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.47 - 1.26i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.25 - 1.05i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.38 + 5.86i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.120 + 0.680i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.53 + 2.13i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.38 + 7.83i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.92 + 5.07i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.22 - 2.71i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.54 + 2.38i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.67 - 9.50i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.78T + 53T^{2} \) |
| 59 | \( 1 + (-2.68 - 0.977i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.07 - 11.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.998 + 0.837i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 1.93i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.05 + 12.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.0 - 9.23i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (11.5 + 9.71i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.14 + 3.72i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.50 + 2.36i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.96433403977515159554712543606, −10.04853339871581898765629291807, −9.299822587112353220302111986966, −7.64191681894801516832035698867, −7.23990552279763373930439975604, −6.05357116062631525462742923227, −4.95407288958511845158909420968, −4.23030973481063829917276791597, −2.34042493034724826339323709632, −0.35245083753449299527304263760,
1.88778373970870550069749574328, 3.58815842613209377771572079360, 5.06548700353235737291568111118, 5.63343790285445914337779015289, 6.64719325623164872648287753605, 7.80156007629524189488594793649, 8.712306335370425292356033178550, 9.922350612018479811869960233322, 10.68282905810921905726699314202, 11.37530890146519543998066277731
