| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 125·5-s + 216·6-s + 836.·7-s + 512·8-s + 729·9-s − 1.00e3·10-s − 2.85e3·11-s + 1.72e3·12-s − 272.·13-s + 6.69e3·14-s − 3.37e3·15-s + 4.09e3·16-s − 1.89e4·17-s + 5.83e3·18-s + 6.85e3·19-s − 8.00e3·20-s + 2.25e4·21-s − 2.28e4·22-s − 8.35e4·23-s + 1.38e4·24-s + 1.56e4·25-s − 2.17e3·26-s + 1.96e4·27-s + 5.35e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.922·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.645·11-s + 0.288·12-s − 0.0343·13-s + 0.652·14-s − 0.258·15-s + 0.250·16-s − 0.937·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.532·21-s − 0.456·22-s − 1.43·23-s + 0.204·24-s + 0.199·25-s − 0.0242·26-s + 0.192·27-s + 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 5 | \( 1 + 125T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 7 | \( 1 - 836.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.85e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 272.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.89e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 8.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.92e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.09e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.42e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.28e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.97e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.38e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.70e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.11e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.07e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.31e6T + 8.07e13T^{2} \) |
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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
−8.993738568754599361169466124074, −8.101009003852621565591612745475, −7.52853688171231056066278650855, −6.45391145876748504887885651466, −5.24761184101300703248539972530, −4.47192048773607028348689079261, −3.57532871182441955531562138569, −2.43999183218334796916715669065, −1.58686644838972866615834414066, 0,
1.58686644838972866615834414066, 2.43999183218334796916715669065, 3.57532871182441955531562138569, 4.47192048773607028348689079261, 5.24761184101300703248539972530, 6.45391145876748504887885651466, 7.52853688171231056066278650855, 8.101009003852621565591612745475, 8.993738568754599361169466124074
