| L(s) = 1 | + 2.35i·3-s − 5.25i·7-s + 3.47·9-s + 19.9i·11-s − 8.47·13-s − 11.8·17-s − 15.2i·19-s + 12.3·21-s − 0.555i·23-s + 29.3i·27-s + 10.9·29-s + 8.29i·31-s − 46.8·33-s − 18.3·37-s − 19.9i·39-s + ⋯ |
| L(s) = 1 | + 0.783i·3-s − 0.751i·7-s + 0.385·9-s + 1.81i·11-s − 0.651·13-s − 0.699·17-s − 0.800i·19-s + 0.588·21-s − 0.0241i·23-s + 1.08i·27-s + 0.377·29-s + 0.267i·31-s − 1.41·33-s − 0.496·37-s − 0.510i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7662881666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7662881666\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.35iT - 9T^{2} \) |
| 7 | \( 1 + 5.25iT - 49T^{2} \) |
| 11 | \( 1 - 19.9iT - 121T^{2} \) |
| 13 | \( 1 + 8.47T + 169T^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 19 | \( 1 + 15.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.555iT - 529T^{2} \) |
| 29 | \( 1 - 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 + 18.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 50.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.55T + 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.8T + 9.40e3T^{2} \) |
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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
−9.551287381084875740101137655241, −9.220762109130813460362303747925, −7.910488878712980847737763079439, −7.11116657325697074700418179951, −6.70347508539105588925702268138, −5.13805522167956291913472608437, −4.56273687643384221154050211214, −4.02064209391968839935089040463, −2.69021185588717532850643582857, −1.52521380817802879652720387713,
0.20114186201946827236426542749, 1.49936157965615445180394907132, 2.54945999870333537207725655915, 3.52972240445450820249983736981, 4.73476446006088263842519538141, 5.79061884826948201452481681353, 6.31625634133989496762791023903, 7.22296769250619144262566982989, 8.114767829331011049040132048454, 8.635708361746120693779569899977
