L(s) = 1 | + (−1 + 1.41i)3-s + (−2.23 − 1.41i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s − 2.82i·13-s − 4·17-s + 6.32i·19-s + (4.23 − 1.74i)21-s − 6.32i·23-s + (5.00 + 1.41i)27-s − 5.65i·29-s + 6.32i·31-s + (−4.00 − 2.82i)33-s + 8.94·37-s + (4.00 + 2.82i)39-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (−0.845 − 0.534i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s − 0.784i·13-s − 0.970·17-s + 1.45i·19-s + (0.924 − 0.381i)21-s − 1.31i·23-s + (0.962 + 0.272i)27-s − 1.05i·29-s + 1.13i·31-s + (−0.696 − 0.492i)33-s + 1.47·37-s + (0.640 + 0.452i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.029353640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029353640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.32iT - 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 6.32iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−9.431181506468021807004243727840, −8.449804265814252085594418434269, −7.58199664248325543507245567480, −6.51530727164020920118128785748, −6.14952809283308466116788426607, −5.03463999652227522960404022742, −4.26499380482963040898820947108, −3.56622722195596193060140454625, −2.42269581131135541615673977431, −0.64197211265481564987827084951,
0.69804681015368937655032499766, 2.14776310040993072740362499879, 2.96691476679884335674545205561, 4.22424790489457997169987651325, 5.28620258211837422733964094934, 6.02717641917944624838808871033, 6.65888768370926168419940143200, 7.29683903228457811063489060674, 8.270656019853471651513624945868, 9.136332945418346558047004302669