| L(s) = 1 | + (−1 − i)3-s + (−3 − 4i)5-s + (7 − 7i)7-s − 7i·9-s − 10·11-s + (9 + 9i)13-s + (−1 + 7i)15-s + (1 − i)17-s − 8i·19-s − 14·21-s + (23 + 23i)23-s + (−7 + 24i)25-s + (−16 + 16i)27-s + 8i·29-s + 14·31-s + ⋯ |
| L(s) = 1 | + (−0.333 − 0.333i)3-s + (−0.600 − 0.800i)5-s + (1 − i)7-s − 0.777i·9-s − 0.909·11-s + (0.692 + 0.692i)13-s + (−0.0666 + 0.466i)15-s + (0.0588 − 0.0588i)17-s − 0.421i·19-s − 0.666·21-s + (1 + i)23-s + (−0.280 + 0.959i)25-s + (−0.592 + 0.592i)27-s + 0.275i·29-s + 0.451·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.784607 - 0.717042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784607 - 0.717042i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3 + 4i)T \) |
| good | 3 | \( 1 + (1 + i)T + 9iT^{2} \) |
| 7 | \( 1 + (-7 + 7i)T - 49iT^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + (-9 - 9i)T + 169iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 289iT^{2} \) |
| 19 | \( 1 + 8iT - 361T^{2} \) |
| 23 | \( 1 + (-23 - 23i)T + 529iT^{2} \) |
| 29 | \( 1 - 8iT - 841T^{2} \) |
| 31 | \( 1 - 14T + 961T^{2} \) |
| 37 | \( 1 + (-33 + 33i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-15 - 15i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-39 + 39i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (7 + 7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 56iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-7 + 7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 98T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49 - 49i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 96iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-63 - 63i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 112iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33 + 33i)T - 9.40e3iT^{2} \) |
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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
−13.70581060962508907695002622572, −12.82899397838247588189098809071, −11.61267478054577667802908693450, −10.93679536380474073344733669899, −9.258201237141827765267687687275, −8.039314837497924889158595047933, −7.02509083252818418422582478892, −5.25112700489574498169748808917, −3.96768690193895358555679991101, −1.03379330716452026642521066076,
2.67840653776458321694065104163, 4.69938114670416991454337161149, 5.89049525383107242513596017353, 7.70398896063801852529497805595, 8.477263128820631142131290060685, 10.39645971781342152745902948694, 11.00550566286039645583661249222, 11.98974020683862037235565252480, 13.33540187767374254163350378370, 14.67535077214901872783188278350
