| L(s) = 1 | − 14·3-s + 64·5-s − 49·7-s − 47·9-s + 420·11-s − 860·13-s − 896·15-s + 830·17-s − 490·19-s + 686·21-s + 4.87e3·23-s + 971·25-s + 4.06e3·27-s − 8.75e3·29-s + 5.62e3·31-s − 5.88e3·33-s − 3.13e3·35-s − 1.43e3·37-s + 1.20e4·39-s − 9.25e3·41-s − 1.47e4·43-s − 3.00e3·45-s − 1.01e4·47-s + 2.40e3·49-s − 1.16e4·51-s + 2.30e4·53-s + 2.68e4·55-s + ⋯ |
| L(s) = 1 | − 0.898·3-s + 1.14·5-s − 0.377·7-s − 0.193·9-s + 1.04·11-s − 1.41·13-s − 1.02·15-s + 0.696·17-s − 0.311·19-s + 0.339·21-s + 1.92·23-s + 0.310·25-s + 1.07·27-s − 1.93·29-s + 1.05·31-s − 0.939·33-s − 0.432·35-s − 0.172·37-s + 1.26·39-s − 0.860·41-s − 1.21·43-s − 0.221·45-s − 0.667·47-s + 1/7·49-s − 0.625·51-s + 1.12·53-s + 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.600004264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600004264\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 14 T + p^{5} T^{2} \) |
| 5 | \( 1 - 64 T + p^{5} T^{2} \) |
| 11 | \( 1 - 420 T + p^{5} T^{2} \) |
| 13 | \( 1 + 860 T + p^{5} T^{2} \) |
| 17 | \( 1 - 830 T + p^{5} T^{2} \) |
| 19 | \( 1 + 490 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4872 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8754 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5628 T + p^{5} T^{2} \) |
| 37 | \( 1 + 1434 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9258 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14756 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10108 T + p^{5} T^{2} \) |
| 53 | \( 1 - 23058 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13734 T + p^{5} T^{2} \) |
| 61 | \( 1 - 25352 T + p^{5} T^{2} \) |
| 67 | \( 1 + 19768 T + p^{5} T^{2} \) |
| 71 | \( 1 - 1792 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37914 T + p^{5} T^{2} \) |
| 79 | \( 1 - 95984 T + p^{5} T^{2} \) |
| 83 | \( 1 + 88242 T + p^{5} T^{2} \) |
| 89 | \( 1 - 43762 T + p^{5} T^{2} \) |
| 97 | \( 1 - 65790 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15611066505934963278239529964, −9.607982489016311644357912149022, −8.725331139872886971425240363561, −7.17644574405736792132237837709, −6.47242915070707926890583131135, −5.55426431366230190913735661326, −4.88734056408431119415537706408, −3.26842077629973548244442876014, −1.98332785044187994223008463440, −0.68030326380338157979972773996,
0.68030326380338157979972773996, 1.98332785044187994223008463440, 3.26842077629973548244442876014, 4.88734056408431119415537706408, 5.55426431366230190913735661326, 6.47242915070707926890583131135, 7.17644574405736792132237837709, 8.725331139872886971425240363561, 9.607982489016311644357912149022, 10.15611066505934963278239529964
