| L(s) = 1 | + (10.7 + 18.6i)2-s + (25.2 − 43.7i)4-s + (−22.4 + 38.8i)5-s + (−4.67e3 − 8.09e3i)7-s + 1.20e4·8-s − 963.·10-s + (−2.75e4 − 4.77e4i)11-s + (3.99e4 − 6.92e4i)13-s + (1.00e5 − 1.73e5i)14-s + (1.16e5 + 2.02e5i)16-s − 8.59e3·17-s + 6.43e5·19-s + (1.13e3 + 1.95e3i)20-s + (5.92e5 − 1.02e6i)22-s + (2.66e5 − 4.61e5i)23-s + ⋯ |
| L(s) = 1 | + (0.474 + 0.822i)2-s + (0.0492 − 0.0853i)4-s + (−0.0160 + 0.0277i)5-s + (−0.735 − 1.27i)7-s + 1.04·8-s − 0.0304·10-s + (−0.567 − 0.983i)11-s + (0.388 − 0.672i)13-s + (0.698 − 1.20i)14-s + (0.445 + 0.772i)16-s − 0.0249·17-s + 1.13·19-s + (0.00158 + 0.00273i)20-s + (0.539 − 0.933i)22-s + (0.198 − 0.344i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.00953 - 0.709481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00953 - 0.709481i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-10.7 - 18.6i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (22.4 - 38.8i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (4.67e3 + 8.09e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.75e4 + 4.77e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-3.99e4 + 6.92e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + 8.59e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-2.66e5 + 4.61e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.64e6 + 4.57e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-1.67e6 + 2.89e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 2.03e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.59e7 - 2.75e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-2.27e6 - 3.94e6i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (9.00e6 + 1.55e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 - 6.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-3.24e7 + 5.61e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.37e7 + 7.58e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (8.37e7 - 1.44e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.55e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.05e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-1.79e8 - 3.11e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.31e8 - 4.01e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 2.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-7.54e7 - 1.30e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−15.24203352693356493223817045057, −13.74923148946231796779112915007, −13.26697390700338365534037434099, −11.07292281892549653936579374851, −10.02026144308974979161357511151, −7.898205111047886432596665188719, −6.70072489850657843783891825745, −5.35443132609567981955017909683, −3.51944119359807174001818600389, −0.77441685556153652769118659155,
1.97980317493199522515047592948, 3.30562952702218813259280676233, 5.14689298681386600952823577846, 7.08403327084708695165025975525, 8.946420819910926147241881410077, 10.39280801996009503667309354392, 11.94491132689474314216972368132, 12.52157154887584082047391017518, 13.80246278635013788084841678447, 15.45752236071765688798573024880
