| L(s) = 1 | + (627. + 362. i)2-s + (1.73e4 + 5.64e4i)3-s + (2.62e5 + 4.54e5i)4-s + (4.33e6 − 2.50e6i)5-s + (−9.56e6 + 4.16e7i)6-s + (1.32e8 − 2.29e8i)7-s + 3.79e8i·8-s + (−2.88e9 + 1.95e9i)9-s + 3.62e9·10-s + (1.04e10 + 6.00e9i)11-s + (−2.10e10 + 2.26e10i)12-s + (4.28e10 + 7.41e10i)13-s + (1.66e11 − 9.58e10i)14-s + (2.16e11 + 2.01e11i)15-s + (−1.37e11 + 2.38e11i)16-s + 1.99e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.293 + 0.955i)3-s + (0.249 + 0.433i)4-s + (0.444 − 0.256i)5-s + (−0.158 + 0.689i)6-s + (0.468 − 0.811i)7-s + 0.353i·8-s + (−0.827 + 0.561i)9-s + 0.362·10-s + (0.401 + 0.231i)11-s + (−0.340 + 0.366i)12-s + (0.310 + 0.537i)13-s + (0.574 − 0.331i)14-s + (0.375 + 0.349i)15-s + (−0.125 + 0.216i)16-s + 0.991i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(3.779842534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.779842534\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-627. - 362. i)T \) |
| 3 | \( 1 + (-1.73e4 - 5.64e4i)T \) |
| good | 5 | \( 1 + (-4.33e6 + 2.50e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-1.32e8 + 2.29e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-1.04e10 - 6.00e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (-4.28e10 - 7.41e10i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 - 1.99e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 1.07e13T + 3.75e25T^{2} \) |
| 23 | \( 1 + (2.69e13 - 1.55e13i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-2.27e14 - 1.31e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-6.33e13 - 1.09e14i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 + 3.75e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (-7.23e15 + 4.17e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (-5.86e15 + 1.01e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (3.98e16 + 2.30e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 - 2.16e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (3.12e17 - 1.80e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (5.34e17 - 9.25e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-1.19e18 - 2.07e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 4.62e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 2.25e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-6.54e18 + 1.13e19i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-9.10e18 - 5.25e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 + 5.46e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-5.11e19 + 8.85e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−14.28020389084791769138844162621, −13.62531338097458079056558148231, −11.77149064412663004607367792273, −10.40633945659762007559059747626, −9.059999511392558614746664922689, −7.54889425009638919432836825312, −5.75734908563770409818050706158, −4.49927516419536137645294614198, −3.45939272735287753992290578897, −1.56449623170073121489864062079,
0.833345393332243599952964322097, 2.12462597024100979444904888311, 3.17066397318044686992083591022, 5.30702821681712635632639036239, 6.43018719993126588191466814113, 8.004850254524111590389719386649, 9.547561732671699380223573423610, 11.42973503161823562854946863965, 12.27405072204350853340173470348, 13.70356404250754811367652464648
