| L(s) = 1 | + 2i·13-s + 6i·17-s + 4·19-s − 8i·23-s − 2·29-s + 4·31-s − 10i·37-s − 2·41-s − 4i·43-s + 8i·47-s + 7·49-s + 2i·53-s + 8·59-s − 2·61-s + 12i·67-s + ⋯ |
| L(s) = 1 | + 0.554i·13-s + 1.45i·17-s + 0.917·19-s − 1.66i·23-s − 0.371·29-s + 0.718·31-s − 1.64i·37-s − 0.312·41-s − 0.609i·43-s + 1.16i·47-s + 49-s + 0.274i·53-s + 1.04·59-s − 0.256·61-s + 1.46i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.959626924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.959626924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.923932089792062189652073376034, −7.33225346048779419575682220050, −6.50046226698284977619425210193, −5.97733570521886341192426614439, −5.17147657418041847102297016203, −4.27845317651902692622930911042, −3.77215898518822284684546202926, −2.70112079267375059466299038404, −1.91414046544617496136066687496, −0.808547064373555332679309238201,
0.64565281169397465826476935697, 1.64767714289116075478053708119, 2.85445161857989997352965804364, 3.30910408569666628188173587970, 4.32759556527766450302584513995, 5.25613984364538224159506795073, 5.51450596492615430898516928190, 6.60218657023607180623347603209, 7.22148408476620081880608888230, 7.80167951805214487445448134445
