| L(s) = 1 | + i·3-s + (2 − i)5-s + 2i·7-s − 9-s − 6·11-s + 2i·13-s + (1 + 2i)15-s + 6i·17-s − 4·19-s − 2·21-s + 8i·23-s + (3 − 4i)25-s − i·27-s + 8·31-s − 6i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s − 1.80·11-s + 0.554i·13-s + (0.258 + 0.516i)15-s + 1.45i·17-s − 0.917·19-s − 0.436·21-s + 1.66i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s + 1.43·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.675033 + 1.09222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.675033 + 1.09222i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.10843074733092975834423502054, −9.677742935051225656092075022861, −8.455995937978732418884194336711, −8.277596149553374920068614923576, −6.72118002178376890734929170176, −5.68519514885472871063497048270, −5.28413821514658668311735752097, −4.18954719814143513552709798199, −2.80070665029808225874580405786, −1.85377158794925610007949732812,
0.55855792703317023892837593089, 2.35021042146029618209966128923, 2.91049022848173918235475093563, 4.62606190514689230156777348903, 5.45498881352701941400877597056, 6.44912600717094385911057205493, 7.18008565536287803926850182462, 7.968926470881624564798648519485, 8.824488093944440135428863578877, 10.16301143950663709372774734815
