| L(s) = 1 | + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (−0.838 − 0.427i)5-s + (−2.47 + 0.594i)7-s + (−0.453 − 0.891i)8-s + (0.290 − 0.894i)10-s + (−0.250 + 0.293i)11-s + (−2.31 − 3.77i)13-s + (−0.974 − 2.35i)14-s + (0.809 − 0.587i)16-s + (5.14 − 0.405i)17-s + (4.21 − 6.88i)19-s + (0.929 + 0.147i)20-s + (−0.329 − 0.201i)22-s + (2.68 + 1.95i)23-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (−0.475 + 0.154i)4-s + (−0.374 − 0.190i)5-s + (−0.936 + 0.224i)7-s + (−0.160 − 0.315i)8-s + (0.0919 − 0.282i)10-s + (−0.0756 + 0.0885i)11-s + (−0.641 − 1.04i)13-s + (−0.260 − 0.628i)14-s + (0.202 − 0.146i)16-s + (1.24 − 0.0982i)17-s + (0.967 − 1.57i)19-s + (0.207 + 0.0329i)20-s + (−0.0701 − 0.0430i)22-s + (0.559 + 0.406i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.773714 - 0.379376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.773714 - 0.379376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (5.09 + 3.87i)T \) |
| good | 5 | \( 1 + (0.838 + 0.427i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (2.47 - 0.594i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (0.250 - 0.293i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.31 + 3.77i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-5.14 + 0.405i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 6.88i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-2.68 - 1.95i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.76 + 0.139i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (1.48 + 0.481i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.21 + 9.88i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (1.87 - 0.297i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.33 + 5.55i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.743 - 9.45i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-6.62 + 9.12i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.14 + 7.25i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-4.12 - 4.82i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (4.10 + 3.50i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (4.00 - 4.00i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.84 - 14.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 8.68iT - 83T^{2} \) |
| 89 | \( 1 + (-2.40 - 10.0i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.93 + 3.35i)T + (15.1 - 95.8i)T^{2} \) |
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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
−9.927927044462605597614529428838, −9.442564231142052184652315903712, −8.424010961210856503999009916090, −7.47750484119310255576693871105, −6.94456298130104112187922444066, −5.63249148891908605121873396664, −5.13951499819334341962941708473, −3.71956594899397586550235537354, −2.82048791878206953651278588573, −0.44462629762674678817542460715,
1.50620216613298540484518807095, 3.12744292687943149619238585656, 3.69357446575919288855363294691, 4.95307539945639689618085535936, 6.00472738800834480586955042494, 7.07840195454868633001356264684, 7.900484872666756571404628779701, 9.034125542798441916613011232615, 9.990389717244496863410086125827, 10.18867503932391091635517221599
