| L(s) = 1 | + (−14.0 + 16.7i)2-s + (−38.4 − 217. i)4-s + (−373. + 1.02e3i)5-s + (−91.6 + 519. i)7-s + (−659. − 380. i)8-s + (−1.19e4 − 2.06e4i)10-s + (2.92e3 + 8.03e3i)11-s + (−2.76e4 + 2.31e4i)13-s + (−7.40e3 − 8.83e3i)14-s + (6.88e4 − 2.50e4i)16-s + (−9.04e4 + 5.22e4i)17-s + (−5.02e4 + 8.71e4i)19-s + (2.37e5 + 4.18e4i)20-s + (−1.75e5 − 6.38e4i)22-s + (−3.58e4 + 6.32e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.877 + 1.04i)2-s + (−0.150 − 0.850i)4-s + (−0.596 + 1.64i)5-s + (−0.0381 + 0.216i)7-s + (−0.160 − 0.0929i)8-s + (−1.19 − 2.06i)10-s + (0.199 + 0.548i)11-s + (−0.966 + 0.811i)13-s + (−0.192 − 0.229i)14-s + (1.05 − 0.382i)16-s + (−1.08 + 0.625i)17-s + (−0.385 + 0.668i)19-s + (1.48 + 0.261i)20-s + (−0.748 − 0.272i)22-s + (−0.128 + 0.0226i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.337759 - 0.117378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.337759 - 0.117378i\) |
| \(L(5)\) |
|
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 | \( 1 + (14.0 - 16.7i)T + (-44.4 - 252. i)T^{2} \) |
| 5 | \( 1 + (373. - 1.02e3i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (91.6 - 519. i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-2.92e3 - 8.03e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (2.76e4 - 2.31e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (9.04e4 - 5.22e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.02e4 - 8.71e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.58e4 - 6.32e3i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (6.37e5 - 7.60e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (1.81e5 + 1.03e6i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-1.04e6 - 1.80e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-5.27e5 - 6.28e5i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-2.50e6 + 9.12e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-8.34e6 - 1.47e6i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 - 5.29e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-5.70e6 + 1.56e7i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (1.10e5 - 6.24e5i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-1.39e7 + 1.17e7i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (1.69e7 - 9.76e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.76e7 - 3.06e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.71e7 - 2.28e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-3.47e7 + 4.14e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (2.57e7 + 1.48e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-8.11e7 + 2.95e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−14.29651939360316398652390849140, −12.45043032521661001801405819008, −11.27874179408633172256377167115, −10.16391487307941816679624103374, −9.079421866196514128519270203668, −7.69774968055476450420074654349, −7.01586138692293955301541779012, −6.14261760170720886346441721552, −3.98791408827109681143831048686, −2.35239820711185662681905684326,
0.21007953167463042306836693702, 0.78311627405923296134633952433, 2.38000659553628796132504774013, 4.11149127065707569653667485349, 5.44140886049615132797335258069, 7.60533359107765838591026328623, 8.762873068577680250021165417937, 9.305534271879264782407175251619, 10.66291027753676219009780741043, 11.69129003811342745096947716693
