| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 125·5-s − 216·6-s + 1.80e3·7-s − 512·8-s + 729·9-s + 1.00e3·10-s − 6.71e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 1.44e4·14-s − 3.37e3·15-s + 4.09e3·16-s − 3.47e4·17-s − 5.83e3·18-s − 1.74e4·19-s − 8.00e3·20-s + 4.88e4·21-s + 5.37e4·22-s + 7.44e4·23-s − 1.38e4·24-s + 1.56e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 1.15e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.99·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.52·11-s + 0.288·12-s + 0.277·13-s − 1.40·14-s − 0.258·15-s + 0.250·16-s − 1.71·17-s − 0.235·18-s − 0.582·19-s − 0.223·20-s + 1.15·21-s + 1.07·22-s + 1.27·23-s − 0.204·24-s + 0.199·25-s − 0.196·26-s + 0.192·27-s + 0.996·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.058764659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058764659\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 125T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 7 | \( 1 - 1.80e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.71e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.47e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.74e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.91e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.31e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.72e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.29e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.84e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.34e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.78e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.11e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.88e6T + 8.07e13T^{2} \) |
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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
−10.25441090472124212883839450104, −8.807293390187757248977616511046, −8.322887663663537748676153500268, −7.74966945332414190343104922761, −6.72123888946138557177447599762, −5.07947308732669513788747130464, −4.41818337738055553481353198688, −2.73141530930218037426250924125, −1.93588307609605089082271684642, −0.71981489524509914712082610984,
0.71981489524509914712082610984, 1.93588307609605089082271684642, 2.73141530930218037426250924125, 4.41818337738055553481353198688, 5.07947308732669513788747130464, 6.72123888946138557177447599762, 7.74966945332414190343104922761, 8.322887663663537748676153500268, 8.807293390187757248977616511046, 10.25441090472124212883839450104
