| L(s) = 1 | + (−8.85 − 15.3i)2-s + (99.0 − 171. i)4-s + (−369. + 640. i)5-s + (3.01e3 + 5.21e3i)7-s − 1.25e4·8-s + 1.31e4·10-s + (9.21e3 + 1.59e4i)11-s + (−5.87e4 + 1.01e5i)13-s + (5.33e4 − 9.24e4i)14-s + (6.07e4 + 1.05e5i)16-s + 4.68e5·17-s + 8.34e5·19-s + (7.32e4 + 1.26e5i)20-s + (1.63e5 − 2.82e5i)22-s + (6.60e5 − 1.14e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.391 − 0.678i)2-s + (0.193 − 0.335i)4-s + (−0.264 + 0.458i)5-s + (0.474 + 0.821i)7-s − 1.08·8-s + 0.414·10-s + (0.189 + 0.328i)11-s + (−0.570 + 0.988i)13-s + (0.371 − 0.643i)14-s + (0.231 + 0.401i)16-s + 1.35·17-s + 1.46·19-s + (0.102 + 0.177i)20-s + (0.148 − 0.257i)22-s + (0.491 − 0.852i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.40009 + 0.153413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40009 + 0.153413i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (8.85 + 15.3i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (369. - 640. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-3.01e3 - 5.21e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-9.21e3 - 1.59e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (5.87e4 - 1.01e5i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 - 4.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-6.60e5 + 1.14e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.21e4 - 5.56e4i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (3.70e6 - 6.42e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 3.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.26e7 - 2.19e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.21e7 + 2.10e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-3.51e6 - 6.09e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 - 3.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (7.68e7 - 1.33e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.75e7 - 1.34e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-6.98e6 + 1.21e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 3.45e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.79e8 + 3.11e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (5.25e7 + 9.09e7i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + 8.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (4.74e7 + 8.21e7i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.06645067847082426823153819490, −14.36863781122622129776152655698, −12.17856633179526119423805323098, −11.52378204455743195965102643231, −10.12762349780123709032516119906, −8.964923920962457073071139516006, −7.09468157138465523840265013163, −5.33687446980248385051579125893, −2.96531326718541365241961219601, −1.43091060991931844259458803025,
0.73630050224835261408036939197, 3.38726190279151930960893307018, 5.42850657126555907442663262285, 7.35767499698485490670148084079, 8.082042883302807490642941218328, 9.716532309241305617856594828549, 11.44306947286797598080751729097, 12.62235294742258191176559403420, 14.18195437839355747337385777085, 15.46727178809138460170976558644
