| L(s) = 1 | + (−1.41 + 0.102i)2-s + (1.97 − 0.289i)4-s − 2.90·5-s + i·7-s + (−2.76 + 0.612i)8-s + (4.10 − 0.298i)10-s − 1.28i·11-s + 1.86i·13-s + (−0.102 − 1.41i)14-s + (3.83 − 1.14i)16-s − 3.87i·17-s − 2.04·19-s + (−5.75 + 0.843i)20-s + (0.132 + 1.81i)22-s + 0.934·23-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0726i)2-s + (0.989 − 0.144i)4-s − 1.30·5-s + 0.377i·7-s + (−0.976 + 0.216i)8-s + (1.29 − 0.0945i)10-s − 0.388i·11-s + 0.518i·13-s + (−0.0274 − 0.376i)14-s + (0.957 − 0.286i)16-s − 0.939i·17-s − 0.468·19-s + (−1.28 + 0.188i)20-s + (0.0282 + 0.387i)22-s + 0.194·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6633311764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6633311764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.102i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 + 1.28iT - 11T^{2} \) |
| 13 | \( 1 - 1.86iT - 13T^{2} \) |
| 17 | \( 1 + 3.87iT - 17T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.934T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 + 1.85iT - 31T^{2} \) |
| 37 | \( 1 + 3.85iT - 37T^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 10.9iT - 59T^{2} \) |
| 61 | \( 1 - 4.70iT - 61T^{2} \) |
| 67 | \( 1 - 4.41T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 16.4iT - 79T^{2} \) |
| 83 | \( 1 - 3.72iT - 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−9.217670763769888760580744388955, −8.793793614596355833598204129750, −7.87213090566892379890458682712, −7.37654389697165466081914237757, −6.52088884158750428263895932096, −5.54663287657917112392562256750, −4.34388327464164792338660239425, −3.31869098085599699478376445143, −2.26722547323350704048038396086, −0.67416112604076515771563885031,
0.63119778316889933272217491795, 2.08999223302712306036579455836, 3.43190152600551529486965608402, 4.07718681616716501872951863883, 5.40197123763996707760059224523, 6.57516653649745849306659364758, 7.24841706415719990875479963545, 7.998866333488354796250480478222, 8.448190492101019112713686718749, 9.392451221441853424323018976787
