L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 125·5-s − 216·6-s + 206.·7-s + 512·8-s + 729·9-s + 1.00e3·10-s − 4.75e3·11-s − 1.72e3·12-s − 4.40e3·13-s + 1.65e3·14-s − 3.37e3·15-s + 4.09e3·16-s + 1.55e4·17-s + 5.83e3·18-s + 6.85e3·19-s + 8.00e3·20-s − 5.58e3·21-s − 3.80e4·22-s + 3.86e4·23-s − 1.38e4·24-s + 1.56e4·25-s − 3.52e4·26-s − 1.96e4·27-s + 1.32e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.227·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.07·11-s − 0.288·12-s − 0.555·13-s + 0.161·14-s − 0.258·15-s + 0.250·16-s + 0.768·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.131·21-s − 0.762·22-s + 0.663·23-s − 0.204·24-s + 0.199·25-s − 0.392·26-s − 0.192·27-s + 0.113·28-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 | \( 1 - 206.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.40e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.55e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 3.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.35e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.48e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.68e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.19e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.35e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.01e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.87e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.36e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.60e6T + 8.07e13T^{2} \) |
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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
−9.390500817103105938497545081819, −8.013877819113456122586975463730, −7.26577554683350355950156980510, −6.24672068484094918018102286802, −5.28568884354088420674156660131, −4.89623264499621618867579917702, −3.49335427051432063136586913920, −2.44859865232308545145693404211, −1.34399828320283444445759141762, 0,
1.34399828320283444445759141762, 2.44859865232308545145693404211, 3.49335427051432063136586913920, 4.89623264499621618867579917702, 5.28568884354088420674156660131, 6.24672068484094918018102286802, 7.26577554683350355950156980510, 8.013877819113456122586975463730, 9.390500817103105938497545081819