| L(s) = 1 | − 2.61·2-s − 3-s + 4.81·4-s + 2.61·6-s − 7-s − 7.34·8-s + 9-s − 4.73·11-s − 4.81·12-s − 13-s + 2.61·14-s + 9.55·16-s + 5.22·17-s − 2.61·18-s + 2.92·19-s + 21-s + 12.3·22-s − 3.33·23-s + 7.34·24-s + 2.61·26-s − 27-s − 4.81·28-s − 0.922·29-s − 7.51·31-s − 10.2·32-s + 4.73·33-s − 13.6·34-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.40·4-s + 1.06·6-s − 0.377·7-s − 2.59·8-s + 0.333·9-s − 1.42·11-s − 1.38·12-s − 0.277·13-s + 0.697·14-s + 2.38·16-s + 1.26·17-s − 0.615·18-s + 0.670·19-s + 0.218·21-s + 2.63·22-s − 0.694·23-s + 1.49·24-s + 0.511·26-s − 0.192·27-s − 0.909·28-s − 0.171·29-s − 1.35·31-s − 1.81·32-s + 0.824·33-s − 2.33·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 + 0.922T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 0.154T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−7.85154702254506232092747004994, −7.21590889472909635967929939662, −6.46507988563227229090388729243, −5.63971272897136047611339106591, −5.17777409086307531692665007848, −3.70262640882965190000898562839, −2.79621608301281199676603820199, −1.99176659221867831715904578705, −0.906226939728232026591100592178, 0,
0.906226939728232026591100592178, 1.99176659221867831715904578705, 2.79621608301281199676603820199, 3.70262640882965190000898562839, 5.17777409086307531692665007848, 5.63971272897136047611339106591, 6.46507988563227229090388729243, 7.21590889472909635967929939662, 7.85154702254506232092747004994
