| L(s) = 1 | − 7.35·2-s + 9·3-s + 22.0·4-s − 66.1·6-s − 251.·7-s + 72.9·8-s + 81·9-s + 121·11-s + 198.·12-s + 633.·13-s + 1.85e3·14-s − 1.24e3·16-s − 460.·17-s − 595.·18-s − 1.69e3·19-s − 2.26e3·21-s − 889.·22-s + 3.13e3·23-s + 656.·24-s − 4.66e3·26-s + 729·27-s − 5.56e3·28-s + 223.·29-s − 7.39e3·31-s + 6.80e3·32-s + 1.08e3·33-s + 3.38e3·34-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.689·4-s − 0.750·6-s − 1.94·7-s + 0.403·8-s + 0.333·9-s + 0.301·11-s + 0.398·12-s + 1.04·13-s + 2.52·14-s − 1.21·16-s − 0.386·17-s − 0.433·18-s − 1.07·19-s − 1.12·21-s − 0.391·22-s + 1.23·23-s + 0.232·24-s − 1.35·26-s + 0.192·27-s − 1.34·28-s + 0.0492·29-s − 1.38·31-s + 1.17·32-s + 0.174·33-s + 0.502·34-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 | \( 1 + 7.35T + 32T^{2} \) |
| 7 | \( 1 + 251.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 633.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 460.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.69e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.13e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 223.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.93e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.50e4T + 8.58e9T^{2} \) |
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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
−9.013471746919159614727375896895, −8.657891556468862949649919457381, −7.38445000719747015045643575331, −6.77706989582179076356349914628, −5.93399835247613943823000104421, −4.22580938517385918341588016687, −3.36685248971807676564235058852, −2.27456759373651103897620473352, −0.992829614442473411000210559796, 0,
0.992829614442473411000210559796, 2.27456759373651103897620473352, 3.36685248971807676564235058852, 4.22580938517385918341588016687, 5.93399835247613943823000104421, 6.77706989582179076356349914628, 7.38445000719747015045643575331, 8.657891556468862949649919457381, 9.013471746919159614727375896895
