L(s) = 1 | + (2.39 + 4.14i)2-s + (−1.75 + 4.88i)3-s + (−7.46 + 12.9i)4-s + (−24.4 + 4.41i)6-s + (15.3 + 26.5i)7-s − 33.1·8-s + (−20.8 − 17.1i)9-s + (−21.3 − 37.0i)11-s + (−50.1 − 59.2i)12-s + (−26.7 + 46.2i)13-s + (−73.4 + 127. i)14-s + (−19.7 − 34.2i)16-s + 0.609·17-s + (21.4 − 127. i)18-s + 94.5·19-s + ⋯ |
L(s) = 1 | + (0.846 + 1.46i)2-s + (−0.338 + 0.941i)3-s + (−0.933 + 1.61i)4-s + (−1.66 + 0.300i)6-s + (0.828 + 1.43i)7-s − 1.46·8-s + (−0.771 − 0.636i)9-s + (−0.586 − 1.01i)11-s + (−1.20 − 1.42i)12-s + (−0.570 + 0.987i)13-s + (−1.40 + 2.42i)14-s + (−0.308 − 0.534i)16-s + 0.00869·17-s + (0.281 − 1.66i)18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.07211 - 1.84043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07211 - 1.84043i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.75 - 4.88i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.39 - 4.14i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-15.3 - 26.5i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (21.3 + 37.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (26.7 - 46.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 0.609T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-76.3 + 132. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-71.6 - 124. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (18.6 - 32.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 274.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-136. + 236. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (92.6 + 160. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-219. - 379. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (248. - 430. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-104. - 181. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (483. - 837. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 401.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 97.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (566. + 980. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-286. - 496. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 90.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-4.62 - 8.00i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.41241563602292394403390625943, −11.73075932989225824122232546249, −10.60908435462872191579315579606, −8.928208266119909284341933360374, −8.619076452767813443802727323249, −7.20256756089981609728042647007, −5.90003403394278509465776872013, −5.29986358060820949634491494215, −4.52284421232525555832975686944, −2.97035046134707321596930954404,
0.72095419653774454072534841764, 1.80591717915197857707219164112, 3.18910015020223006389328658757, 4.68714450643323500483703313724, 5.38460642124170740754106691714, 7.25205155453510858454501433353, 7.81637752030287208205364014741, 9.818435166206320912232763415825, 10.53865921299619143810508693105, 11.35419459128035190326110580834