| L(s) = 1 | + (6.15 + 10.6i)2-s + (40.5 + 23.3i)3-s + (52.1 − 90.3i)4-s + (599. − 346. i)5-s + 575. i·6-s + (−419. − 2.36e3i)7-s + 4.43e3·8-s + (1.09e3 + 1.89e3i)9-s + (7.38e3 + 4.26e3i)10-s + (−3.93e3 + 6.82e3i)11-s + (4.22e3 − 2.43e3i)12-s + 1.73e4i·13-s + (2.26e4 − 1.90e4i)14-s + 3.23e4·15-s + (1.39e4 + 2.42e4i)16-s + (7.12e4 + 4.11e4i)17-s + ⋯ |
| L(s) = 1 | + (0.384 + 0.666i)2-s + (0.5 + 0.288i)3-s + (0.203 − 0.352i)4-s + (0.959 − 0.554i)5-s + 0.444i·6-s + (−0.174 − 0.984i)7-s + 1.08·8-s + (0.166 + 0.288i)9-s + (0.738 + 0.426i)10-s + (−0.268 + 0.465i)11-s + (0.203 − 0.117i)12-s + 0.608i·13-s + (0.589 − 0.495i)14-s + 0.639·15-s + (0.213 + 0.369i)16-s + (0.853 + 0.492i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.89615 + 0.440923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.89615 + 0.440923i\) |
| \(L(5)\) |
|
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 | \( 1 + (419. + 2.36e3i)T \) |
| good | 2 | \( 1 + (-6.15 - 10.6i)T + (-128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-599. + 346. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (3.93e3 - 6.82e3i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.73e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-7.12e4 - 4.11e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.78e4 - 3.34e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.31e5 + 2.27e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 9.41e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (6.65e5 + 3.84e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.75e6 - 3.03e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 2.49e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.71e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.24e6 - 1.87e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (2.12e6 - 3.68e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (8.47e6 + 4.89e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.00e7 - 1.15e7i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (7.18e6 - 1.24e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.47e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (4.43e7 + 2.55e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.40e6 - 2.42e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.15e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-9.41e7 + 5.43e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 4.54e7iT - 7.83e15T^{2} \) |
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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
−16.40806496087550095195574017306, −14.88655355417528844350878378842, −13.97244065344417827767076822586, −12.97441765734404569347730186113, −10.58256412466351246454011097906, −9.555424053293439495739688102546, −7.58875352722203607003870277056, −5.98791016066219051696019324869, −4.40033372889625938006431509199, −1.65927664671602808410626134671,
2.07818564801305892094284391600, 3.17952572251982675574973689158, 5.80086593336898999305589357751, 7.69231403259516071786148643733, 9.465404179018445673063650084252, 10.98620621451551029087622339614, 12.44875541436144020789541105034, 13.41374984279092770560946724089, 14.61795650884522057437454049095, 16.15673731399667066982765311571
