| L(s) = 1 | − 0.801·3-s + 2.80i·5-s + 2.69i·7-s − 2.35·9-s + 1.19i·11-s − 2.24i·15-s − 1.13·17-s + 1.93i·19-s − 2.15i·21-s − 4.60·23-s − 2.85·25-s + 4.29·27-s − 7.89·29-s + 5.89i·31-s − 0.960i·33-s + ⋯ |
| L(s) = 1 | − 0.462·3-s + 1.25i·5-s + 1.01i·7-s − 0.785·9-s + 0.361i·11-s − 0.580i·15-s − 0.275·17-s + 0.444i·19-s − 0.471i·21-s − 0.959·23-s − 0.570·25-s + 0.826·27-s − 1.46·29-s + 1.05i·31-s − 0.167i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4290541061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4290541061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.801T + 3T^{2} \) |
| 5 | \( 1 - 2.80iT - 5T^{2} \) |
| 7 | \( 1 - 2.69iT - 7T^{2} \) |
| 11 | \( 1 - 1.19iT - 11T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 1.93iT - 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 - 5.89iT - 31T^{2} \) |
| 37 | \( 1 - 0.951iT - 37T^{2} \) |
| 41 | \( 1 + 3.31iT - 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 - 7.69iT - 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 - 0.0120iT - 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 + 9.25iT - 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 0.807T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 14.7iT - 89T^{2} \) |
| 97 | \( 1 + 3.13iT - 97T^{2} \) |
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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
−9.269871307609507449740371753596, −8.621897782933711997817992655171, −7.68226564481593691482252506460, −6.99164598597475051673924411488, −5.96712759408738436391867512425, −5.85541323501805077793668960039, −4.71867831282440675094683073189, −3.54225100841430409835749217595, −2.74298772545006303586897203845, −1.93042924712915159414699905062,
0.16409153340135742236739215088, 1.05592478915976210761685630646, 2.39963553934197007333555456459, 3.79483620084616979302901160719, 4.36109758089093135056752433391, 5.35792647971785701007579892626, 5.82322123255568854837122643666, 6.82294917722149886339658378389, 7.70655194517745680825838710719, 8.375834877940717778515086041323
