| L(s) = 1 | + 4.38·5-s − i·7-s − 5.11i·11-s + 5.21i·13-s − 2i·17-s + 6.50·19-s + 4.63·23-s + 14.2·25-s + 1.26·29-s + 4.21i·31-s − 4.38i·35-s + 1.98i·37-s + 7.37i·41-s − 3.71·43-s − 2.57·47-s + ⋯ |
| L(s) = 1 | + 1.96·5-s − 0.377i·7-s − 1.54i·11-s + 1.44i·13-s − 0.485i·17-s + 1.49·19-s + 0.967·23-s + 2.84·25-s + 0.235·29-s + 0.756i·31-s − 0.741i·35-s + 0.326i·37-s + 1.15i·41-s − 0.566·43-s − 0.375·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383481849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383481849\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 4.38T + 5T^{2} \) |
| 11 | \( 1 + 5.11iT - 11T^{2} \) |
| 13 | \( 1 - 5.21iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 4.21iT - 31T^{2} \) |
| 37 | \( 1 - 1.98iT - 37T^{2} \) |
| 41 | \( 1 - 7.37iT - 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 3.50iT - 59T^{2} \) |
| 61 | \( 1 - 4.53iT - 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 3.98iT - 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−8.154544375056189311259929437458, −7.07457854321502980076018781784, −6.54912281716281854738371855206, −5.98144210143069802449799107554, −5.20445894896794421484320520582, −4.70105123484283757082564850647, −3.29190441893512916146232250979, −2.83316748512958257491657654022, −1.64063992526293053522789000613, −1.03921693088365904270909527807,
1.07084058956806449728231798869, 1.95921825406442540299731609109, 2.61910101344009414843902186909, 3.46202422481423322843058727736, 4.98389784020655718023100551903, 5.13895521074228364535627318514, 5.90537441518223936409468573739, 6.58758616077691832326637896308, 7.32958186308090921697791044990, 8.072465236859452047361771064275
