| L(s) = 1 | + 17·3-s + 25·5-s + 49·7-s + 46·9-s + 715·11-s + 331·13-s + 425·15-s − 1.69e3·17-s + 1.71e3·19-s + 833·21-s + 3.95e3·23-s + 625·25-s − 3.34e3·27-s + 4.57e3·29-s − 6.75e3·31-s + 1.21e4·33-s + 1.22e3·35-s − 1.65e4·37-s + 5.62e3·39-s + 1.88e4·41-s − 2.25e3·43-s + 1.15e3·45-s + 537·47-s + 2.40e3·49-s − 2.88e4·51-s − 1.09e4·53-s + 1.78e4·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 0.447·5-s + 0.377·7-s + 0.189·9-s + 1.78·11-s + 0.543·13-s + 0.487·15-s − 1.42·17-s + 1.09·19-s + 0.412·21-s + 1.55·23-s + 1/5·25-s − 0.884·27-s + 1.01·29-s − 1.26·31-s + 1.94·33-s + 0.169·35-s − 1.98·37-s + 0.592·39-s + 1.75·41-s − 0.186·43-s + 0.0846·45-s + 0.0354·47-s + 1/7·49-s − 1.55·51-s − 0.537·53-s + 0.796·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.439674905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.439674905\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 17 T + p^{5} T^{2} \) |
| 11 | \( 1 - 65 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 331 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1699 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1718 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3950 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4579 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6756 T + p^{5} T^{2} \) |
| 37 | \( 1 + 16518 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18876 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2258 T + p^{5} T^{2} \) |
| 47 | \( 1 - 537 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10984 T + p^{5} T^{2} \) |
| 59 | \( 1 - 25956 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39188 T + p^{5} T^{2} \) |
| 67 | \( 1 + 4416 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31880 T + p^{5} T^{2} \) |
| 73 | \( 1 + 5018 T + p^{5} T^{2} \) |
| 79 | \( 1 - 27977 T + p^{5} T^{2} \) |
| 83 | \( 1 + 37644 T + p^{5} T^{2} \) |
| 89 | \( 1 + 17216 T + p^{5} T^{2} \) |
| 97 | \( 1 + 63175 T + p^{5} T^{2} \) |
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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
−9.616341752204721234981467613159, −8.931159329516103129174263039296, −8.620976051471550281204503311598, −7.25411754803954508900552603364, −6.54993863678859625147566664642, −5.31304616648520207312814377671, −4.08638594092850549990173093219, −3.20783728009924229394418555822, −2.03842815551172372641556271846, −1.06711003462642782928815437541,
1.06711003462642782928815437541, 2.03842815551172372641556271846, 3.20783728009924229394418555822, 4.08638594092850549990173093219, 5.31304616648520207312814377671, 6.54993863678859625147566664642, 7.25411754803954508900552603364, 8.620976051471550281204503311598, 8.931159329516103129174263039296, 9.616341752204721234981467613159
