| L(s) = 1 | − 7.38·2-s + 9·3-s + 22.4·4-s − 66.4·6-s + 191.·7-s + 70.3·8-s + 81·9-s − 121·11-s + 202.·12-s − 505.·13-s − 1.41e3·14-s − 1.23e3·16-s + 1.77e3·17-s − 597.·18-s + 528.·19-s + 1.72e3·21-s + 893.·22-s − 70.7·23-s + 632.·24-s + 3.72e3·26-s + 729·27-s + 4.30e3·28-s − 3.23e3·29-s − 3.63e3·31-s + 6.88e3·32-s − 1.08e3·33-s − 1.31e4·34-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.577·3-s + 0.702·4-s − 0.753·6-s + 1.47·7-s + 0.388·8-s + 0.333·9-s − 0.301·11-s + 0.405·12-s − 0.829·13-s − 1.92·14-s − 1.20·16-s + 1.49·17-s − 0.434·18-s + 0.335·19-s + 0.852·21-s + 0.393·22-s − 0.0278·23-s + 0.224·24-s + 1.08·26-s + 0.192·27-s + 1.03·28-s − 0.715·29-s − 0.680·31-s + 1.18·32-s − 0.174·33-s − 1.94·34-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 | \( 1 + 7.38T + 32T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 505.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.77e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 528.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 70.7T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.92e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.83e4T + 8.58e9T^{2} \) |
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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
−8.947912253690755025342205104226, −8.172513028001128961988790407883, −7.72777057436860229686546079152, −7.04504585947027281815879118767, −5.36593523954015362624985042258, −4.67864647719384203089808859913, −3.29991823749229860349175253152, −1.92834995802977983806163871024, −1.35754642285161192595027909557, 0,
1.35754642285161192595027909557, 1.92834995802977983806163871024, 3.29991823749229860349175253152, 4.67864647719384203089808859913, 5.36593523954015362624985042258, 7.04504585947027281815879118767, 7.72777057436860229686546079152, 8.172513028001128961988790407883, 8.947912253690755025342205104226
