| L(s) = 1 | + 9.15·2-s + 9·3-s + 51.7·4-s + 82.3·6-s + 209.·7-s + 180.·8-s + 81·9-s − 121·11-s + 465.·12-s + 335.·13-s + 1.91e3·14-s − 1.19·16-s + 799.·17-s + 741.·18-s − 658.·19-s + 1.88e3·21-s − 1.10e3·22-s + 4.11e3·23-s + 1.62e3·24-s + 3.07e3·26-s + 729·27-s + 1.08e4·28-s + 559.·29-s − 6.05e3·31-s − 5.79e3·32-s − 1.08e3·33-s + 7.31e3·34-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.577·3-s + 1.61·4-s + 0.934·6-s + 1.61·7-s + 0.999·8-s + 0.333·9-s − 0.301·11-s + 0.933·12-s + 0.551·13-s + 2.61·14-s − 0.00117·16-s + 0.670·17-s + 0.539·18-s − 0.418·19-s + 0.933·21-s − 0.487·22-s + 1.62·23-s + 0.576·24-s + 0.891·26-s + 0.192·27-s + 2.61·28-s + 0.123·29-s − 1.13·31-s − 1.00·32-s − 0.174·33-s + 1.08·34-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
| \(L(3)\) |
\(\approx\) |
\(9.967128159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.967128159\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 9.15T + 32T^{2} \) |
| 7 | \( 1 - 209.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 335.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 799.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 658.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 559.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.58e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.61e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.58e9T^{2} \) |
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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
−9.399472366885101029866247140743, −8.403279613105015977521709977681, −7.68323922743350348139970946069, −6.74529481194748802660006530881, −5.59280426523777388045727780825, −4.93930270400051264063442955535, −4.16980003219276927072116143095, −3.20498895299337076467000052134, −2.22773869420500529429195118287, −1.21608535605291894008587321271,
1.21608535605291894008587321271, 2.22773869420500529429195118287, 3.20498895299337076467000052134, 4.16980003219276927072116143095, 4.93930270400051264063442955535, 5.59280426523777388045727780825, 6.74529481194748802660006530881, 7.68323922743350348139970946069, 8.403279613105015977521709977681, 9.399472366885101029866247140743
