| L(s) = 1 | − 4.46i·5-s + 2.64·7-s + 4.46i·11-s − 7.34i·17-s − 8.64·19-s + 2.88i·23-s − 14.9·25-s − 8.29·31-s − 11.8i·35-s − 3.93·37-s − 2.88i·41-s + 7.00·49-s + 19.9·55-s − 10.2i·71-s + 11.8i·77-s + ⋯ |
| L(s) = 1 | − 1.99i·5-s + 0.999·7-s + 1.34i·11-s − 1.78i·17-s − 1.98·19-s + 0.601i·23-s − 2.98·25-s − 1.48·31-s − 1.99i·35-s − 0.647·37-s − 0.450i·41-s + 49-s + 2.68·55-s − 1.21i·71-s + 1.34i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8972819903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8972819903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 4.46iT - 5T^{2} \) |
| 11 | \( 1 - 4.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 + 8.64T + 19T^{2} \) |
| 23 | \( 1 - 2.88iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 2.88iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.467042656768785697382189044367, −7.67104865278592566441094040829, −7.08743612439311620998034489680, −5.77322219928957286700049287249, −4.98863158800812318920988707241, −4.67573949519844408537769600837, −3.93958572650166429128345566939, −2.14658754433232075816287952598, −1.59564993816268375051735941066, −0.25751857565385617252215911255,
1.79070178846138437458027219830, 2.53810728575792409064889053264, 3.62489446604934505372337612264, 4.09404699640716328795613414980, 5.52446079690529930841310761948, 6.24085836357607020114012344346, 6.66388610303957964213708399664, 7.65959884368767755289767480168, 8.291680676290566892053322968197, 8.843376895432560156434243785953
