| L(s) = 1 | + 17.3·2-s + 27·3-s + 174.·4-s + 113.·5-s + 469.·6-s − 892.·7-s + 807.·8-s + 729·9-s + 1.96e3·10-s + 4.70e3·12-s − 5.26e3·13-s − 1.55e4·14-s + 3.05e3·15-s − 8.28e3·16-s − 1.58e4·17-s + 1.26e4·18-s + 7.01e3·19-s + 1.97e4·20-s − 2.40e4·21-s − 4.30e4·23-s + 2.18e4·24-s − 6.53e4·25-s − 9.14e4·26-s + 1.96e4·27-s − 1.55e5·28-s + 4.68e4·29-s + 5.30e4·30-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.577·3-s + 1.36·4-s + 0.404·5-s + 0.887·6-s − 0.983·7-s + 0.557·8-s + 0.333·9-s + 0.621·10-s + 0.786·12-s − 0.664·13-s − 1.51·14-s + 0.233·15-s − 0.505·16-s − 0.784·17-s + 0.512·18-s + 0.234·19-s + 0.551·20-s − 0.567·21-s − 0.737·23-s + 0.321·24-s − 0.836·25-s − 1.02·26-s + 0.192·27-s − 1.33·28-s + 0.357·29-s + 0.358·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 17.3T + 128T^{2} \) |
| 5 | \( 1 - 113.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 892.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 5.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.58e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 7.01e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.68e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.00e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.87e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.97e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.16e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.94e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.43e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.94e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.68e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.746135773448339026289280154660, −9.025060958335319141727663779159, −7.58413831203773442320737600920, −6.58093360139717017537559996594, −5.84976680089324320775165341898, −4.71405537293907907931220711165, −3.76739989752191574490017737419, −2.85407234131552376817161469222, −1.98705768537749381887813943256, 0,
1.98705768537749381887813943256, 2.85407234131552376817161469222, 3.76739989752191574490017737419, 4.71405537293907907931220711165, 5.84976680089324320775165341898, 6.58093360139717017537559996594, 7.58413831203773442320737600920, 9.025060958335319141727663779159, 9.746135773448339026289280154660
