| L(s) = 1 | − 0.622i·3-s + 0.657i·5-s − 2.30i·7-s + 2.61·9-s − 4.95i·11-s + 5.87·13-s + 0.409·15-s + (2.82 + 2.99i)17-s + 1.65·19-s − 1.43·21-s + 4.54i·23-s + 4.56·25-s − 3.49i·27-s + 3.32i·29-s − 2.90i·31-s + ⋯ |
| L(s) = 1 | − 0.359i·3-s + 0.294i·5-s − 0.869i·7-s + 0.870·9-s − 1.49i·11-s + 1.63·13-s + 0.105·15-s + (0.686 + 0.727i)17-s + 0.380·19-s − 0.312·21-s + 0.947i·23-s + 0.913·25-s − 0.672i·27-s + 0.616i·29-s − 0.521i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484211976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484211976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-2.82 - 2.99i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.622iT - 3T^{2} \) |
| 5 | \( 1 - 0.657iT - 5T^{2} \) |
| 7 | \( 1 + 2.30iT - 7T^{2} \) |
| 11 | \( 1 + 4.95iT - 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 - 4.54iT - 23T^{2} \) |
| 29 | \( 1 - 3.32iT - 29T^{2} \) |
| 31 | \( 1 + 2.90iT - 31T^{2} \) |
| 37 | \( 1 + 0.0987iT - 37T^{2} \) |
| 41 | \( 1 - 5.90iT - 41T^{2} \) |
| 43 | \( 1 - 9.96T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 + 3.44iT - 71T^{2} \) |
| 73 | \( 1 - 0.343iT - 73T^{2} \) |
| 79 | \( 1 + 4.40iT - 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 5.04iT - 97T^{2} \) |
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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
−8.201608016417890135940734512395, −7.67441075774190623504281247892, −6.87248257400792940515806602454, −6.17581637475121872805667929402, −5.62123700670856967339613106379, −4.38576005280331734076723256352, −3.60854445070125132408801541318, −3.11345801787711789420075884541, −1.43514723243924464518270105602, −0.970502368836808952431028048006,
1.08654348342737901158428572947, 2.04541720067676410791550638796, 3.11581454576422889347836441254, 4.08472635921629157479460538930, 4.77305687327611318465997532703, 5.42366613721607291640191219719, 6.38120270244240828087915003486, 7.03636346872015482059392321839, 7.86535473913812206583227169874, 8.635148308267443538338571921522
