| L(s) = 1 | + (2.13 + 2.54i)2-s + (−2.03 − 2.20i)3-s + (−1.22 + 6.94i)4-s + (−2.35 − 6.46i)5-s + (1.26 − 9.89i)6-s + (1.10 + 6.28i)7-s + (−8.77 + 5.06i)8-s + (−0.712 + 8.97i)9-s + (11.4 − 19.8i)10-s + (0.425 − 1.16i)11-s + (17.7 − 11.4i)12-s + (−4.09 − 3.43i)13-s + (−13.6 + 16.2i)14-s + (−9.45 + 18.3i)15-s + (−5.16 − 1.88i)16-s + (−13.7 − 7.93i)17-s + ⋯ |
| L(s) = 1 | + (1.06 + 1.27i)2-s + (−0.678 − 0.734i)3-s + (−0.306 + 1.73i)4-s + (−0.470 − 1.29i)5-s + (0.210 − 1.64i)6-s + (0.158 + 0.897i)7-s + (−1.09 + 0.633i)8-s + (−0.0791 + 0.996i)9-s + (1.14 − 1.98i)10-s + (0.0386 − 0.106i)11-s + (1.48 − 0.952i)12-s + (−0.314 − 0.264i)13-s + (−0.973 + 1.16i)14-s + (−0.630 + 1.22i)15-s + (−0.322 − 0.117i)16-s + (−0.808 − 0.466i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09151 + 0.526803i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09151 + 0.526803i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.03 + 2.20i)T \) |
| good | 2 | \( 1 + (-2.13 - 2.54i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (2.35 + 6.46i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 6.28i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.425 + 1.16i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (4.09 + 3.43i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (13.7 + 7.93i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.78 - 11.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-24.4 - 4.30i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (19.9 + 23.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.75 - 15.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-26.0 + 45.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (0.694 - 0.827i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (45.3 + 16.5i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (56.7 - 10.0i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 13.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.6 - 56.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-17.5 - 99.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (60.2 + 50.5i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-39.7 - 22.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (34.4 + 59.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-30.2 + 25.4i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (27.1 + 32.3i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (61.8 - 35.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (1.06 + 0.387i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−16.84434209571865059676080952519, −16.12793982966632406526382484094, −15.02684893095847823618306811374, −13.42895458122139200987325800431, −12.62079219807375605633348510253, −11.70594348614743024447362950447, −8.660828027875335962308235470774, −7.41947612242851705274742992356, −5.77422188087595539728686421331, −4.79939024460854675355573544713,
3.37327000446252900178412925693, 4.70226332963684578534888474253, 6.79023250489525286899507629207, 9.938013315144268327231371228054, 11.02881173150562032285243428306, 11.40516646100668612490836999467, 13.05554981765594584651435951044, 14.46721383524602835632250187981, 15.22034587079311406514517158293, 16.98183817314100644394643061117
