| L(s) = 1 | + (−0.557 − 1.29i)2-s + (−1.37 + 1.44i)4-s + (1.27 − 3.49i)5-s + (1.63 + 0.287i)7-s + (2.65 + 0.986i)8-s + (−5.24 + 0.293i)10-s + (0.526 − 0.191i)11-s + (1.99 − 1.66i)13-s + (−0.534 − 2.28i)14-s + (−0.195 − 3.99i)16-s + (−4.50 + 2.60i)17-s + (−0.925 − 0.534i)19-s + (3.30 + 6.65i)20-s + (−0.542 − 0.577i)22-s + (−1.54 − 8.77i)23-s + ⋯ |
| L(s) = 1 | + (−0.393 − 0.919i)2-s + (−0.689 + 0.724i)4-s + (0.568 − 1.56i)5-s + (0.616 + 0.108i)7-s + (0.937 + 0.348i)8-s + (−1.65 + 0.0927i)10-s + (0.158 − 0.0577i)11-s + (0.551 − 0.463i)13-s + (−0.142 − 0.609i)14-s + (−0.0487 − 0.998i)16-s + (−1.09 + 0.631i)17-s + (−0.212 − 0.122i)19-s + (0.739 + 1.48i)20-s + (−0.115 − 0.123i)22-s + (−0.322 − 1.82i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.550731 - 0.999163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.550731 - 0.999163i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.557 + 1.29i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.27 + 3.49i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.526 + 0.191i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.99 + 1.66i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.50 - 2.60i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.925 + 0.534i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.54 + 8.77i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.702 + 0.837i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 0.559i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 5.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.556 - 0.662i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.690 - 1.89i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.22 - 6.95i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 6.80iT - 53T^{2} \) |
| 59 | \( 1 + (-8.57 - 3.11i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.832 + 4.71i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.06 - 8.42i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.98 - 6.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.92 + 3.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.28 + 7.49i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.28 + 1.08i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.9 - 6.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.74 + 1.36i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.27841555579803065896368030326, −10.38782218561446458974170302774, −9.391480557146649042219204745625, −8.471250438990435291194844197163, −8.225356958638801465838160035854, −6.26100417590858803654084841112, −4.89847720554087748878680212902, −4.23611324758729659526326088108, −2.30987433647104281112074649086, −1.02981696797142138647592615543,
1.99981138724671247257869018988, 3.81635476368993639162618388381, 5.27503267470443952702785863496, 6.38526476355687359598136904101, 6.99722012771753396830033489538, 7.945349663140308374137348886001, 9.149286661083566224473025284005, 9.946940926073941290255095531241, 10.92397680340233133207245715442, 11.48375577010571479113591046874
