L(s) = 1 | + 1.29·2-s − 9·3-s − 30.3·4-s − 11.6·6-s + 91.7·7-s − 80.5·8-s + 81·9-s − 121·11-s + 272.·12-s + 70.3·13-s + 118.·14-s + 866.·16-s − 1.17e3·17-s + 104.·18-s − 961.·19-s − 825.·21-s − 156.·22-s + 1.30e3·23-s + 725.·24-s + 90.9·26-s − 729·27-s − 2.78e3·28-s − 3.34e3·29-s + 6.45e3·31-s + 3.69e3·32-s + 1.08e3·33-s − 1.51e3·34-s + ⋯ |
L(s) = 1 | + 0.228·2-s − 0.577·3-s − 0.947·4-s − 0.131·6-s + 0.707·7-s − 0.445·8-s + 0.333·9-s − 0.301·11-s + 0.547·12-s + 0.115·13-s + 0.161·14-s + 0.846·16-s − 0.986·17-s + 0.0761·18-s − 0.611·19-s − 0.408·21-s − 0.0688·22-s + 0.513·23-s + 0.256·24-s + 0.0263·26-s − 0.192·27-s − 0.670·28-s − 0.738·29-s + 1.20·31-s + 0.638·32-s + 0.174·33-s − 0.225·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 1.29T + 32T^{2} \) |
| 7 | \( 1 - 91.7T + 1.68e4T^{2} \) |
| 13 | \( 1 - 70.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 961.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.34e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.56e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.26e5T + 8.58e9T^{2} \) |
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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
−8.987147247642206333077671845823, −8.326641919629730322284099110188, −7.34129743003162568563125122351, −6.24515060305776542197248337514, −5.36087053126938948918080143844, −4.61555315168013506998808554603, −3.89553999728919550571677984437, −2.42128174153922966663182708004, −1.05550844884314548849999393766, 0,
1.05550844884314548849999393766, 2.42128174153922966663182708004, 3.89553999728919550571677984437, 4.61555315168013506998808554603, 5.36087053126938948918080143844, 6.24515060305776542197248337514, 7.34129743003162568563125122351, 8.326641919629730322284099110188, 8.987147247642206333077671845823