| L(s) = 1 | + (−2 + 2.23i)3-s + 4i·5-s + 8.94·7-s + (−1.00 − 8.94i)9-s + 4.47i·11-s + 17.8·13-s + (−8.94 − 8i)15-s − 17.8i·17-s + 20·19-s + (−17.8 + 20.0i)21-s − 16i·23-s + 9·25-s + (22.0 + 15.6i)27-s − 52i·29-s + 26.8·31-s + ⋯ |
| L(s) = 1 | + (−0.666 + 0.745i)3-s + 0.800i·5-s + 1.27·7-s + (−0.111 − 0.993i)9-s + 0.406i·11-s + 1.37·13-s + (−0.596 − 0.533i)15-s − 1.05i·17-s + 1.05·19-s + (−0.851 + 0.952i)21-s − 0.695i·23-s + 0.359·25-s + (0.814 + 0.579i)27-s − 1.79i·29-s + 0.865·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.902863985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902863985\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 - 2.23i)T \) |
| good | 5 | \( 1 - 4iT - 25T^{2} \) |
| 7 | \( 1 - 8.94T + 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 - 17.8T + 169T^{2} \) |
| 17 | \( 1 + 17.8iT - 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 + 16iT - 529T^{2} \) |
| 29 | \( 1 + 52iT - 841T^{2} \) |
| 31 | \( 1 - 26.8T + 961T^{2} \) |
| 37 | \( 1 + 53.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 102. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−10.45538772331074819334875743816, −9.525102961940081556643110958433, −8.604949640938713326472870978299, −7.60065012986719340596668078770, −6.65364591297747870710388689786, −5.71677700872871977321131061401, −4.81427986004129152535170575064, −3.95131935146338699545799978255, −2.70522866401765373102466150185, −1.01589874925361876581605265108,
1.07902397741686234209396157843, 1.64308797259786427134807508849, 3.54527821938954884458569596526, 4.92444239602724890589805999927, 5.42500492082428128920810935536, 6.42820932354745428256118555880, 7.48325609181453489630503571363, 8.448580857311919869292734862886, 8.671752818129063617640612958634, 10.25109878667250891782487033919
