L(s) = 1 | + (16.6 − 42.0i)2-s + (−491. + 491. i)3-s + (−1.49e3 − 1.40e3i)4-s + (6.77e3 + 6.77e3i)5-s + (1.24e4 + 2.88e4i)6-s − 4.75e4i·7-s + (−8.38e4 + 3.94e4i)8-s − 3.05e5i·9-s + (3.98e5 − 1.72e5i)10-s + (−3.48e5 − 3.48e5i)11-s + (1.42e6 − 4.41e4i)12-s + (6.23e5 − 6.23e5i)13-s + (−1.99e6 − 7.91e5i)14-s − 6.65e6·15-s + (2.60e5 + 4.18e6i)16-s − 4.47e5·17-s + ⋯ |
L(s) = 1 | + (0.368 − 0.929i)2-s + (−1.16 + 1.16i)3-s + (−0.728 − 0.684i)4-s + (0.969 + 0.969i)5-s + (0.655 + 1.51i)6-s − 1.06i·7-s + (−0.905 + 0.425i)8-s − 1.72i·9-s + (1.25 − 0.544i)10-s + (−0.652 − 0.652i)11-s + (1.64 − 0.0512i)12-s + (0.465 − 0.465i)13-s + (−0.993 − 0.393i)14-s − 2.26·15-s + (0.0620 + 0.998i)16-s − 0.0765·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.645304 - 0.903699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.645304 - 0.903699i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-16.6 + 42.0i)T \) |
good | 3 | \( 1 + (491. - 491. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (-6.77e3 - 6.77e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 + 4.75e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (3.48e5 + 3.48e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (-6.23e5 + 6.23e5i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 + 4.47e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-1.39e7 + 1.39e7i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 + 4.84e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (4.44e6 - 4.44e6i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 1.41e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-4.51e7 - 4.51e7i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 3.91e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-8.76e8 - 8.76e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (3.45e9 + 3.45e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (5.84e9 + 5.84e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (-1.80e9 + 1.80e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (5.33e9 - 5.33e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 - 6.24e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.44e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.55e8T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-5.06e8 + 5.06e8i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 + 4.97e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 5.39e10T + 7.15e21T^{2} \) |
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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
−16.06432009358966742173861150998, −14.45710182294310854393187249513, −13.25058585448565315962464991347, −11.12363738691216640393909946544, −10.67140959627533002234602814880, −9.685875719518262677577454674315, −6.21207639536678586924692750616, −4.87731603301900207555436552154, −3.16275015046054309938154413337, −0.51840656566161467837431182763,
1.57055331308372955021749335559, 5.35737227928772801546051256425, 5.84434742708443330966886422070, 7.56348636299946189714019462971, 9.296917396431104193470013751289, 11.96940256679530788009332124082, 12.76178418346802473433755260392, 13.77240293951549055803548918074, 15.81728216588505480126092896800, 16.92867955768199371601555215408