| L(s) = 1 | + (−14.0 − 6.67i)3-s + 28.8·5-s − 44.6i·7-s + (153. + 188. i)9-s − 152. i·11-s + 578. i·13-s + (−407. − 192. i)15-s − 82.5i·17-s − 1.23e3·19-s + (−297. + 628. i)21-s + 2.59e3·23-s − 2.28e3·25-s + (−914. − 3.67e3i)27-s + 53.2·29-s − 4.03e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.903 − 0.428i)3-s + 0.516·5-s − 0.344i·7-s + (0.633 + 0.773i)9-s − 0.379i·11-s + 0.948i·13-s + (−0.467 − 0.221i)15-s − 0.0692i·17-s − 0.783·19-s + (−0.147 + 0.311i)21-s + 1.02·23-s − 0.732·25-s + (−0.241 − 0.970i)27-s + 0.0117·29-s − 0.753i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5808823768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5808823768\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (14.0 + 6.67i)T \) |
| good | 5 | \( 1 - 28.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 44.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 152. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 578. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 82.5iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 53.2T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.92e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.85e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.01e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.40e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.34e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 3.70e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 3.28e4T + 8.58e9T^{2} \) |
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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
−10.94507629417744216500697696215, −10.01968259067398275829770565751, −9.060724097988825836823213740808, −7.84345276488503083392349352612, −6.82164044151613846871335482377, −6.13375514878225193645765186032, −5.09634018961422161161940683456, −4.02664928320123667629829875076, −2.27770990879763798565350562538, −1.15466613006743241062333281089,
0.17429923864173440330222463672, 1.61391890442363115675088479669, 3.13777867445844293664142438900, 4.52070102863723847842204105928, 5.42516828639542361999861532129, 6.21508088198650650235134663518, 7.22417272276150943394294510754, 8.539523347518253842269354908679, 9.532328446014646944262480000786, 10.32707968503229128989679498170
