| L(s) = 1 | + (−4.65 + 8.06i)2-s + (13.5 − 7.79i)3-s + (−11.3 − 19.6i)4-s + (151. + 87.3i)5-s + 145. i·6-s + (−271. + 209. i)7-s − 384.·8-s + (121.5 − 210. i)9-s + (−1.40e3 + 813. i)10-s + (92.6 + 160. i)11-s + (−305. − 176. i)12-s + 3.98e3i·13-s + (−424. − 3.16e3i)14-s + 2.72e3·15-s + (2.51e3 − 4.35e3i)16-s + (6.10e3 − 3.52e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.581 + 1.00i)2-s + (0.5 − 0.288i)3-s + (−0.176 − 0.306i)4-s + (1.21 + 0.699i)5-s + 0.671i·6-s + (−0.791 + 0.610i)7-s − 0.751·8-s + (0.166 − 0.288i)9-s + (−1.40 + 0.813i)10-s + (0.0696 + 0.120i)11-s + (−0.176 − 0.102i)12-s + 1.81i·13-s + (−0.154 − 1.15i)14-s + 0.807·15-s + (0.614 − 1.06i)16-s + (1.24 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.706460 + 1.23296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706460 + 1.23296i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.5 + 7.79i)T \) |
| 7 | \( 1 + (271. - 209. i)T \) |
| good | 2 | \( 1 + (4.65 - 8.06i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-151. - 87.3i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-92.6 - 160. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.98e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-6.10e3 + 3.52e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.06e3 + 2.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-3.32e3 + 5.75e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.93e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-2.07e4 + 1.19e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.97e4 + 3.42e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 4.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.31e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.29e5 - 7.47e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.09e4 - 8.81e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.31e5 + 7.61e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.45e5 + 8.42e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.05e4 + 5.29e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.85e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (7.73e3 - 4.46e3i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (2.21e5 - 3.84e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 5.59e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.79e5 + 1.03e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.23e6iT - 8.32e11T^{2} \) |
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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
−17.19607885889093517526385603164, −16.14296654351800889972093633895, −14.73608123889513936969192962618, −13.81250444304388767025469934863, −12.12781465633240036265484573496, −9.789403262925515602133770988915, −8.925530265620749234944812959570, −7.01677059026643105071608940861, −6.13910874393800482084465522634, −2.59187709771297644202042665241,
1.12389530671090047301397717855, 3.09802591702431119873927515125, 5.81124299874847215197604180979, 8.462672915140198393392114469279, 9.975446585185686289348671231333, 10.26922981570550744162039026848, 12.51436206579679159008124615531, 13.44657392240594873309610739126, 15.08875210130603258452530979576, 16.75009966213475529745844588161
