| L(s) = 1 | + (−25.4 − 14.7i)2-s + (−7.75e3 − 1.34e4i)4-s + (−6.80e4 + 3.93e4i)5-s + (−5.76e5 + 9.98e5i)7-s + 9.38e5i·8-s + 2.31e6·10-s + (2.03e7 + 1.17e7i)11-s + (6.00e6 + 1.04e7i)13-s + (2.93e7 − 1.69e7i)14-s + (−1.13e8 + 1.96e8i)16-s − 5.59e8i·17-s − 4.54e8·19-s + (1.05e9 + 6.10e8i)20-s + (−3.44e8 − 5.97e8i)22-s + (−2.35e9 + 1.35e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.199 − 0.114i)2-s + (−0.473 − 0.820i)4-s + (−0.871 + 0.503i)5-s + (−0.700 + 1.21i)7-s + 0.447i·8-s + 0.231·10-s + (1.04 + 0.601i)11-s + (0.0957 + 0.165i)13-s + (0.278 − 0.160i)14-s + (−0.422 + 0.731i)16-s − 1.36i·17-s − 0.508·19-s + (0.825 + 0.476i)20-s + (−0.138 − 0.239i)22-s + (−0.690 + 0.398i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.6027072418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6027072418\) |
| \(L(8)\) |
|
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 | \( 1 + (25.4 + 14.7i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (6.80e4 - 3.93e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (5.76e5 - 9.98e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-2.03e7 - 1.17e7i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (-6.00e6 - 1.04e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 + 5.59e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 4.54e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + (2.35e9 - 1.35e9i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-4.85e9 - 2.80e9i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (1.92e10 + 3.33e10i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + 1.53e11T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-1.37e11 + 7.94e10i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-2.50e11 + 4.33e11i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-2.81e11 - 1.62e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + 3.38e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-4.28e12 + 2.47e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (2.12e11 - 3.67e11i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-1.23e12 - 2.14e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 - 7.44e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 4.08e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-1.40e12 + 2.43e12i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-1.18e13 - 6.81e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 - 7.66e12iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-2.12e13 + 3.68e13i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−14.02039974416053566257479320266, −12.29778528001069441888962982210, −11.31777016343891391630022705632, −9.721237024329404044686978344401, −8.860790468741762133061059117458, −6.99267134773735265820719315946, −5.59357827345739772914000079928, −3.91994762490233363044102781065, −2.16801873048279753910257409358, −0.27640027030198872820998667638,
0.866925979374322302901614666318, 3.61662266449338636820922191231, 4.16852386542840118489968978630, 6.58711527209908350457274443741, 7.926506259544264628270360978226, 8.907598467227143027966700019965, 10.51377819401301128795512850685, 12.10100943027843809398809108033, 13.00484798019697668072386026790, 14.23292954083018828986823514797
