| L(s) = 1 | + (0.644 + 0.540i)2-s + (0.739 − 5.14i)3-s + (−1.26 − 7.18i)4-s + (10.1 + 3.69i)5-s + (3.25 − 2.91i)6-s + (−4.63 + 26.2i)7-s + (6.43 − 11.1i)8-s + (−25.9 − 7.60i)9-s + (4.54 + 7.87i)10-s + (17.1 − 6.22i)11-s + (−37.8 + 1.20i)12-s + (−27.1 + 22.7i)13-s + (−17.1 + 14.4i)14-s + (26.5 − 49.5i)15-s + (−44.6 + 16.2i)16-s + (56.4 + 97.8i)17-s + ⋯ |
| L(s) = 1 | + (0.227 + 0.191i)2-s + (0.142 − 0.989i)3-s + (−0.158 − 0.897i)4-s + (0.909 + 0.330i)5-s + (0.221 − 0.198i)6-s + (−0.250 + 1.41i)7-s + (0.284 − 0.492i)8-s + (−0.959 − 0.281i)9-s + (0.143 + 0.249i)10-s + (0.469 − 0.170i)11-s + (−0.911 + 0.0289i)12-s + (−0.578 + 0.485i)13-s + (−0.328 + 0.275i)14-s + (0.456 − 0.852i)15-s + (−0.697 + 0.253i)16-s + (0.805 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.32056 - 0.453148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32056 - 0.453148i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.739 + 5.14i)T \) |
| good | 2 | \( 1 + (-0.644 - 0.540i)T + (1.38 + 7.87i)T^{2} \) |
| 5 | \( 1 + (-10.1 - 3.69i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (4.63 - 26.2i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-17.1 + 6.22i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (27.1 - 22.7i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-56.4 - 97.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-9.33 + 16.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (24.1 + 137. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-61.6 - 51.7i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (42.0 + 238. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (16.7 + 29.0i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (297. - 249. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (79.2 - 28.8i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-80.1 + 454. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 351.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (47.3 + 17.2i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (24.6 - 139. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (79.8 - 67.0i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-161. - 280. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (292. - 506. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (486. + 408. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-740. - 621. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-233. + 403. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.56e3 + 571. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−16.88554678200133468740048887138, −14.99152616658038698811336340778, −14.35824754064933989509947668013, −13.11453715951536114781206193746, −11.89850315260074013003218359751, −10.02461366457003226825794705308, −8.726053700375607856107708513663, −6.49013171186753472735014375868, −5.71386263896642540627790974466, −2.08666146997398087738577047962,
3.44469370570110204412695572750, 5.02499560056153580832387316024, 7.49396701856777052585476221376, 9.305275267733759671334075690395, 10.29385727416662643969242323626, 11.91124368758628129859292720846, 13.52974881548668990364157101514, 14.13852835921140285163427033145, 16.05874410039137570704918106768, 17.01790441042041029220654668973
