| L(s) = 1 | − 3-s + 25·5-s − 49·7-s − 242·9-s + 453·11-s − 969·13-s − 25·15-s + 1.63e3·17-s + 1.55e3·19-s + 49·21-s + 1.65e3·23-s + 625·25-s + 485·27-s − 4.98e3·29-s − 1.19e3·31-s − 453·33-s − 1.22e3·35-s − 1.10e4·37-s + 969·39-s − 1.72e3·41-s + 1.08e4·43-s − 6.05e3·45-s − 2.62e4·47-s + 2.40e3·49-s − 1.63e3·51-s + 2.59e4·53-s + 1.13e4·55-s + ⋯ |
| L(s) = 1 | − 0.0641·3-s + 0.447·5-s − 0.377·7-s − 0.995·9-s + 1.12·11-s − 1.59·13-s − 0.0286·15-s + 1.37·17-s + 0.985·19-s + 0.0242·21-s + 0.651·23-s + 1/5·25-s + 0.128·27-s − 1.10·29-s − 0.222·31-s − 0.0724·33-s − 0.169·35-s − 1.32·37-s + 0.102·39-s − 0.160·41-s + 0.891·43-s − 0.445·45-s − 1.73·47-s + 1/7·49-s − 0.0881·51-s + 1.26·53-s + 0.504·55-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + T + p^{5} T^{2} \) |
| 11 | \( 1 - 453 T + p^{5} T^{2} \) |
| 13 | \( 1 + 969 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1637 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1550 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1654 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4985 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1192 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11018 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1728 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10814 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26237 T + p^{5} T^{2} \) |
| 53 | \( 1 - 25936 T + p^{5} T^{2} \) |
| 59 | \( 1 - 4580 T + p^{5} T^{2} \) |
| 61 | \( 1 + 12488 T + p^{5} T^{2} \) |
| 67 | \( 1 - 15848 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51792 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4846 T + p^{5} T^{2} \) |
| 79 | \( 1 + 62765 T + p^{5} T^{2} \) |
| 83 | \( 1 - 23644 T + p^{5} T^{2} \) |
| 89 | \( 1 + 147300 T + p^{5} T^{2} \) |
| 97 | \( 1 + 8343 T + p^{5} T^{2} \) |
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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
−9.543538046471764630152849894926, −8.880602516006217601798289818688, −7.62554005630913192388611599453, −6.86536183999845443653507492508, −5.71726696045434833088435956042, −5.11201887030831891432761688711, −3.60191736160286866933116757355, −2.71186632685840340404323915358, −1.35176238200643169071789916648, 0,
1.35176238200643169071789916648, 2.71186632685840340404323915358, 3.60191736160286866933116757355, 5.11201887030831891432761688711, 5.71726696045434833088435956042, 6.86536183999845443653507492508, 7.62554005630913192388611599453, 8.880602516006217601798289818688, 9.543538046471764630152849894926
