| L(s) = 1 | + (−15.2 − 26.3i)2-s + (40.5 + 23.3i)3-s + (−334. + 579. i)4-s + (890. − 514. i)5-s − 1.42e3i·6-s + (2.24e3 + 843. i)7-s + 1.25e4·8-s + (1.09e3 + 1.89e3i)9-s + (−2.71e4 − 1.56e4i)10-s + (300. − 521. i)11-s + (−2.71e4 + 1.56e4i)12-s + 7.31e3i·13-s + (−1.19e4 − 7.20e4i)14-s + 4.81e4·15-s + (−1.05e5 − 1.83e5i)16-s + (−1.77e4 − 1.02e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.950 − 1.64i)2-s + (0.5 + 0.288i)3-s + (−1.30 + 2.26i)4-s + (1.42 − 0.822i)5-s − 1.09i·6-s + (0.936 + 0.351i)7-s + 3.07·8-s + (0.166 + 0.288i)9-s + (−2.71 − 1.56i)10-s + (0.0205 − 0.0355i)11-s + (−1.30 + 0.755i)12-s + 0.256i·13-s + (−0.311 − 1.87i)14-s + 0.950·15-s + (−1.61 − 2.79i)16-s + (−0.212 − 0.122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.969225 - 1.22540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969225 - 1.22540i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-40.5 - 23.3i)T \) |
| 7 | \( 1 + (-2.24e3 - 843. i)T \) |
| good | 2 | \( 1 + (15.2 + 26.3i)T + (-128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-890. + 514. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-300. + 521. i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 7.31e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.77e4 + 1.02e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.09e5 + 6.33e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.34e5 + 2.33e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 9.12e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (6.06e5 + 3.50e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-4.63e5 - 8.03e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.59e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.68e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.52e6 - 2.03e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-3.54e6 + 6.14e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-9.80e5 - 5.65e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.12e7 - 6.47e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.72e7 - 2.98e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.76e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.69e7 - 9.80e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.52e7 + 4.37e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 4.65e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (6.94e7 - 4.01e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 5.45e7iT - 7.83e15T^{2} \) |
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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
−16.54312747124270509913710558351, −14.06820298755149978421027907063, −13.03843172422078790224859574823, −11.68517343657367067693130932555, −10.21605053882127744159381797930, −9.235694207240874280754946049102, −8.336134264462456050566410073180, −4.74670573332781030707676964579, −2.46688656610988894716838436107, −1.30359795458164007050240176522,
1.54699572602940162253268319882, 5.46987237234486967055325910137, 6.79856219651705367750977434454, 7.998425791594647860988835609388, 9.454226266632998094424888465519, 10.50554613292663366230522937297, 13.78421859357881840810301300001, 14.20646949323638846876171576863, 15.32438189712423258104845294477, 16.92805845338928116302436143912
