| L(s) = 1 | + 1.73i·3-s − 3.46·7-s − 2.99·9-s + 3.46·11-s − 4i·13-s − 6·17-s + 3.46i·19-s − 5.99i·21-s − 3.46i·23-s − 5.19i·27-s − 6i·29-s + 3.46i·31-s + 5.99i·33-s − 4i·37-s + 6.92·39-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s − 1.30·7-s − 0.999·9-s + 1.04·11-s − 1.10i·13-s − 1.45·17-s + 0.794i·19-s − 1.30i·21-s − 0.722i·23-s − 0.999i·27-s − 1.11i·29-s + 0.622i·31-s + 1.04i·33-s − 0.657i·37-s + 1.10·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5432592857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5432592857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.543164146028374427561550395777, −8.956371372494822665402240988268, −8.167349241715704277579917945953, −6.81892618183809578386247119114, −6.20258216299889506951707518857, −5.31078089506086390221748774585, −4.10560628699450513153379626617, −3.54055089719056972149027694542, −2.44086686353836655021261288553, −0.22997612311721221793524808045,
1.43956885054680820709638343322, 2.64085198424837836323584157817, 3.66685784919888398238459616308, 4.81507585246492848997811675112, 6.24960796384318447227484620682, 6.59848319230140380703242914973, 7.15008703768597228134920931290, 8.395583487761025175244392305063, 9.221911177060172432347406314760, 9.562541694836885040300049481189
