L(s) = 1 | + 3-s + (1 − 2i)5-s − 2i·7-s + 9-s − 2i·11-s + 2·13-s + (1 − 2i)15-s + 4i·17-s + 2i·19-s − 2i·21-s − 6i·23-s + (−3 − 4i)25-s + 27-s − 4i·29-s − 8·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (0.447 − 0.894i)5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s + 0.554·13-s + (0.258 − 0.516i)15-s + 0.970i·17-s + 0.458i·19-s − 0.436i·21-s − 1.25i·23-s + (−0.600 − 0.800i)25-s + 0.192·27-s − 0.742i·29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67343 - 1.20614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67343 - 1.20614i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.927552420453697507127993756516, −8.758514178063275267563581222078, −8.464382178538791126905696001078, −7.49886976770475233595022465125, −6.36361920797486524263920111276, −5.57745285432679839474960107288, −4.32809844973046431951389949443, −3.68196767008157501868807950936, −2.18419089687037272141547525349, −0.946090791114472922337670514859,
1.80107749269515286491746855798, 2.76749875908977418541463166365, 3.64827733511076420208503472582, 5.03893009182737536907298803387, 5.90884910800367383644708730339, 6.96503066497852641375843745291, 7.52506845560616517194788636232, 8.678017027444621528053369816238, 9.432003937323121926259927836612, 9.945390098040302828120174546708