| L(s) = 1 | + (6.09 − 10.5i)2-s + (−10.3 − 17.9i)4-s + (246. + 426. i)5-s + (−382. + 662. i)7-s + 1.30e3·8-s + 6.00e3·10-s + (−36.3 + 62.9i)11-s + (−3.01e3 − 5.21e3i)13-s + (4.66e3 + 8.07e3i)14-s + (9.30e3 − 1.61e4i)16-s + 5.98e3·17-s + 1.86e4·19-s + (5.09e3 − 8.82e3i)20-s + (443. + 767. i)22-s + (1.21e4 + 2.10e4i)23-s + ⋯ |
| L(s) = 1 | + (0.538 − 0.933i)2-s + (−0.0808 − 0.140i)4-s + (0.880 + 1.52i)5-s + (−0.421 + 0.729i)7-s + 0.903·8-s + 1.89·10-s + (−0.00823 + 0.0142i)11-s + (−0.380 − 0.658i)13-s + (0.454 + 0.786i)14-s + (0.567 − 0.983i)16-s + 0.295·17-s + 0.624·19-s + (0.142 − 0.246i)20-s + (0.00887 + 0.0153i)22-s + (0.208 + 0.360i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.56919 + 0.121694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56919 + 0.121694i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-6.09 + 10.5i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-246. - 426. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (382. - 662. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (36.3 - 62.9i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.01e3 + 5.21e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 5.98e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-1.21e4 - 2.10e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.33e4 + 7.51e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.05e5 + 1.83e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.96e5 - 3.39e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.43e5 + 5.95e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-3.20e5 + 5.55e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 8.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.25e6 + 2.18e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.21e5 + 3.83e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.96e5 + 5.12e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.48e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-4.44e5 + 7.70e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.69e6 - 2.93e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 1.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.30e6 + 7.44e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| 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
−15.47090109099223185275043981685, −14.25537147444196182981885367909, −13.21351576627169887733548554314, −11.91936915035843869089970362191, −10.70937673328409706776451751608, −9.728669970311792195417903897872, −7.32998115856567636598876036841, −5.71352875224569826777232806578, −3.26646087005394778022548003631, −2.26434947898053426175958650163,
1.27930418734725997511411275023, 4.54094893402121111555745794736, 5.68666726407582876305044241227, 7.17972809275833330489712328694, 8.956154080752669346267722025083, 10.29128414347433225692940334847, 12.44634186919072924237519657775, 13.52137387267298226028189717889, 14.31280784511177393245219735344, 16.08696678626030997315221765431
