| L(s) = 1 | + 26·3-s − 25·5-s + 22·7-s + 433·9-s + 768·11-s − 46·13-s − 650·15-s + 378·17-s − 1.10e3·19-s + 572·21-s + 1.98e3·23-s + 625·25-s + 4.94e3·27-s − 5.61e3·29-s + 3.98e3·31-s + 1.99e4·33-s − 550·35-s − 142·37-s − 1.19e3·39-s + 1.54e3·41-s + 5.02e3·43-s − 1.08e4·45-s − 2.47e4·47-s − 1.63e4·49-s + 9.82e3·51-s − 1.41e4·53-s − 1.92e4·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.447·5-s + 0.169·7-s + 1.78·9-s + 1.91·11-s − 0.0754·13-s − 0.745·15-s + 0.317·17-s − 0.699·19-s + 0.283·21-s + 0.782·23-s + 1/5·25-s + 1.30·27-s − 1.23·29-s + 0.745·31-s + 3.19·33-s − 0.0758·35-s − 0.0170·37-s − 0.125·39-s + 0.143·41-s + 0.414·43-s − 0.796·45-s − 1.63·47-s − 0.971·49-s + 0.529·51-s − 0.692·53-s − 0.855·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.144861981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.144861981\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 26 T + p^{5} T^{2} \) |
| 7 | \( 1 - 22 T + p^{5} T^{2} \) |
| 11 | \( 1 - 768 T + p^{5} T^{2} \) |
| 13 | \( 1 + 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 378 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1986 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5610 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3988 T + p^{5} T^{2} \) |
| 37 | \( 1 + 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1542 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5026 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24738 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14166 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5522 T + p^{5} T^{2} \) |
| 67 | \( 1 - 24742 T + p^{5} T^{2} \) |
| 71 | \( 1 + 42372 T + p^{5} T^{2} \) |
| 73 | \( 1 + 52126 T + p^{5} T^{2} \) |
| 79 | \( 1 - 39640 T + p^{5} T^{2} \) |
| 83 | \( 1 - 59826 T + p^{5} T^{2} \) |
| 89 | \( 1 - 57690 T + p^{5} T^{2} \) |
| 97 | \( 1 + 144382 T + p^{5} T^{2} \) |
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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
−13.64488512822499785595729067068, −12.51008447626271481984709159140, −11.27305973059014327671936740755, −9.623278127801182954229469221863, −8.876541211307687114492425864116, −7.87412416245702588257884861989, −6.67301845428911318524815128969, −4.30225772078511804630397848926, −3.26554872208621395466809701666, −1.58096832430823094617676330741,
1.58096832430823094617676330741, 3.26554872208621395466809701666, 4.30225772078511804630397848926, 6.67301845428911318524815128969, 7.87412416245702588257884861989, 8.876541211307687114492425864116, 9.623278127801182954229469221863, 11.27305973059014327671936740755, 12.51008447626271481984709159140, 13.64488512822499785595729067068
