| L(s) = 1 | − 1.55i·3-s + (−2.03 + 0.928i)5-s + 1.46·7-s + 0.593·9-s + 2.19i·11-s − 3.35·13-s + (1.44 + 3.15i)15-s − 3.77i·17-s + (−4.27 − 0.851i)19-s − 2.26i·21-s + 7.11·23-s + (3.27 − 3.77i)25-s − 5.57i·27-s − 6.65i·29-s + 1.93·31-s + ⋯ |
| L(s) = 1 | − 0.895i·3-s + (−0.909 + 0.415i)5-s + 0.552·7-s + 0.197·9-s + 0.661i·11-s − 0.931·13-s + (0.372 + 0.814i)15-s − 0.916i·17-s + (−0.980 − 0.195i)19-s − 0.495i·21-s + 1.48·23-s + (0.654 − 0.755i)25-s − 1.07i·27-s − 1.23i·29-s + 0.346·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208944529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208944529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.03 - 0.928i)T \) |
| 19 | \( 1 + (4.27 + 0.851i)T \) |
| good | 3 | \( 1 + 1.55iT - 3T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 2.19iT - 11T^{2} \) |
| 13 | \( 1 + 3.35T + 13T^{2} \) |
| 17 | \( 1 + 3.77iT - 17T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 + 6.65iT - 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 - 4.72T + 37T^{2} \) |
| 41 | \( 1 + 7.88iT - 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 3.80T + 61T^{2} \) |
| 67 | \( 1 + 0.493iT - 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 + 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 0.949T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 0.649T + 97T^{2} \) |
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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
−9.143373027578073201545538131500, −8.166604718376395585573523538859, −7.39068863582212460533981423945, −7.16610007370644960117158902260, −6.23099811219978275796832732756, −4.80502632656531359988899164949, −4.35958931088749207908499930937, −2.89670322849723984942048142466, −2.01581029113973870043303630680, −0.52277526845661624651651874960,
1.28925175777824011607726910756, 2.95560316596776006259717114615, 3.92742825643501138419798693933, 4.65636108053331267183840430105, 5.20992588204956731989139311234, 6.49849966550345111415725187567, 7.44996145450290754760882039771, 8.234696791940134968118404596417, 8.878473829009369579367543889688, 9.638293607082353012905647010514
