| L(s) = 1 | + (1.41 + 1.73i)5-s − 2.82i·7-s − 4.89·11-s − 4.89i·13-s − 3.46i·17-s − 6.92·19-s + 4i·23-s + (−0.999 + 4.89i)25-s − 8.48·29-s − 6.92·31-s + (4.89 − 4.00i)35-s + 4.89i·37-s + 5.65·41-s − 11.3i·43-s − 4i·47-s + ⋯ |
| L(s) = 1 | + (0.632 + 0.774i)5-s − 1.06i·7-s − 1.47·11-s − 1.35i·13-s − 0.840i·17-s − 1.58·19-s + 0.834i·23-s + (−0.199 + 0.979i)25-s − 1.57·29-s − 1.24·31-s + (0.828 − 0.676i)35-s + 0.805i·37-s + 0.883·41-s − 1.72i·43-s − 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6371111584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6371111584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−9.432141449193287430990704623900, −8.243803174576588564866670351633, −7.45154295754985602583817806419, −7.01519159048463812079826917381, −5.74123050512277648622973103999, −5.28099599754929713139865987027, −3.93479528477775695460814942162, −3.01052863371629128375359692377, −2.03443569204560425507151807990, −0.22674511464239499541192414297,
1.91556364174571135073578283438, 2.40427836586881665935847244869, 4.05486508567339992959846704225, 4.89817714344101325132712300850, 5.78615286568700133808980288169, 6.29023639320354471053663068122, 7.55761915840280330769418052071, 8.480284076577217273481340186280, 8.969263069286632818923700895018, 9.654170765050161519704818968140
