L(s) = 1 | + (0.758 + 2.33i)2-s + (−0.809 − 0.587i)3-s + (−3.26 + 2.36i)4-s + (−0.309 + 0.951i)5-s + (0.758 − 2.33i)6-s + (−2.65 + 1.93i)7-s + (−4.03 − 2.93i)8-s + (0.309 + 0.951i)9-s − 2.45·10-s + (2.96 + 1.47i)11-s + 4.03·12-s + (−0.0967 − 0.297i)13-s + (−6.53 − 4.74i)14-s + (0.809 − 0.587i)15-s + (1.29 − 3.98i)16-s + (1.54 − 4.75i)17-s + ⋯ |
L(s) = 1 | + (0.536 + 1.65i)2-s + (−0.467 − 0.339i)3-s + (−1.63 + 1.18i)4-s + (−0.138 + 0.425i)5-s + (0.309 − 0.953i)6-s + (−1.00 + 0.730i)7-s + (−1.42 − 1.03i)8-s + (0.103 + 0.317i)9-s − 0.776·10-s + (0.894 + 0.446i)11-s + 1.16·12-s + (−0.0268 − 0.0825i)13-s + (−1.74 − 1.26i)14-s + (0.208 − 0.151i)15-s + (0.323 − 0.996i)16-s + (0.374 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102390 + 1.05151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102390 + 1.05151i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-2.96 - 1.47i)T \) |
good | 2 | \( 1 + (-0.758 - 2.33i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (2.65 - 1.93i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (0.0967 + 0.297i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 4.75i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.03 - 4.38i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + (-4.07 + 2.96i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.06 - 3.28i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.13 - 1.54i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (8.77 + 6.37i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.52 - 4.69i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.41 + 5.38i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 8.73i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + (-0.949 + 2.92i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.00 + 5.08i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.67 - 5.14i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.02 + 15.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 + (-0.0692 - 0.213i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.66575861978809145214808044198, −12.36284770347997070138270078749, −11.87630336827339338470479782513, −9.981709508665628229338175745660, −8.976052115657130829684758337613, −7.62172752980393347425946717621, −6.82148417837969108039694699746, −6.03144301687022952365354579795, −5.02056400090550295996883660776, −3.38561193733536342553195425618,
0.996802772288967517521983178922, 3.27881105943847527635504645399, 4.09236183625956386239701851532, 5.36491894572619182328363524680, 6.80463306473484993856974795309, 8.832093102505381844901694938200, 9.804841894729485021608432733766, 10.46111032208628832702414573752, 11.55629346904288359917734427243, 12.12861012778694424971077985452