| L(s) = 1 | + 0.517i·5-s + i·7-s + 5.63·11-s + 1.91·13-s − 1.29i·17-s + 4.23i·19-s + 4.38·23-s + 4.73·25-s + 3.70i·29-s + 5.11i·31-s − 0.517·35-s + 2.18·37-s − 1.88i·41-s − 5.37i·43-s − 8.32·47-s + ⋯ |
| L(s) = 1 | + 0.231i·5-s + 0.377i·7-s + 1.69·11-s + 0.531·13-s − 0.314i·17-s + 0.971i·19-s + 0.913·23-s + 0.946·25-s + 0.687i·29-s + 0.918i·31-s − 0.0874·35-s + 0.358·37-s − 0.294i·41-s − 0.820i·43-s − 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.438994664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438994664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.517iT - 5T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 1.29iT - 17T^{2} \) |
| 19 | \( 1 - 4.23iT - 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 - 3.70iT - 29T^{2} \) |
| 31 | \( 1 - 5.11iT - 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 1.88iT - 41T^{2} \) |
| 43 | \( 1 + 5.37iT - 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 5.53iT - 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 0.0617T + 61T^{2} \) |
| 67 | \( 1 + 7.31iT - 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 16.9iT - 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 - 1.33iT - 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−8.264171191602131491254014590472, −7.33935604202371017185813379468, −6.65246019649911300583156583831, −6.22039765387463484286863386864, −5.29569728721254105483534794560, −4.54180134412750086189025089044, −3.59058753528493698231950568948, −3.10338363555786948246002180601, −1.81552149453283991106919097199, −1.05682590600848104032037128752,
0.77063613472039240075296921307, 1.50767556363360682612811344730, 2.73499446850814366572809617864, 3.64917764487856473079897451004, 4.31258612190528184280019376290, 4.97308394999107413175113776547, 6.02147218494099228039500986031, 6.61747530279755952482660097573, 7.10349415907982869246731453004, 8.102654636167769515871886440863
