| L(s) = 1 | + 24·2-s − 1.47e3·4-s − 4.83e3·5-s − 1.67e4·7-s − 8.44e4·8-s − 1.15e5·10-s − 5.34e5·11-s − 5.77e5·13-s − 4.01e5·14-s + 9.87e5·16-s + 6.90e6·17-s + 1.06e7·19-s + 7.10e6·20-s − 1.28e7·22-s − 1.86e7·23-s − 2.54e7·25-s − 1.38e7·26-s + 2.46e7·28-s − 1.28e8·29-s − 5.28e7·31-s + 1.96e8·32-s + 1.65e8·34-s + 8.08e7·35-s − 1.82e8·37-s + 2.55e8·38-s + 4.08e8·40-s − 3.08e8·41-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.718·4-s − 0.691·5-s − 0.376·7-s − 0.911·8-s − 0.366·10-s − 1.00·11-s − 0.431·13-s − 0.199·14-s + 0.235·16-s + 1.17·17-s + 0.987·19-s + 0.496·20-s − 0.530·22-s − 0.603·23-s − 0.522·25-s − 0.228·26-s + 0.270·28-s − 1.16·29-s − 0.331·31-s + 1.03·32-s + 0.625·34-s + 0.260·35-s − 0.431·37-s + 0.523·38-s + 0.630·40-s − 0.415·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3 p^{3} T + p^{11} T^{2} \) |
| 5 | \( 1 + 966 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 2392 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 534612 T + p^{11} T^{2} \) |
| 13 | \( 1 + 577738 T + p^{11} T^{2} \) |
| 17 | \( 1 - 6905934 T + p^{11} T^{2} \) |
| 19 | \( 1 - 10661420 T + p^{11} T^{2} \) |
| 23 | \( 1 + 18643272 T + p^{11} T^{2} \) |
| 29 | \( 1 + 128406630 T + p^{11} T^{2} \) |
| 31 | \( 1 + 52843168 T + p^{11} T^{2} \) |
| 37 | \( 1 + 182213314 T + p^{11} T^{2} \) |
| 41 | \( 1 + 308120442 T + p^{11} T^{2} \) |
| 43 | \( 1 + 17125708 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2687348496 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1596055698 T + p^{11} T^{2} \) |
| 59 | \( 1 - 5189203740 T + p^{11} T^{2} \) |
| 61 | \( 1 - 6956478662 T + p^{11} T^{2} \) |
| 67 | \( 1 + 15481826884 T + p^{11} T^{2} \) |
| 71 | \( 1 + 9791485272 T + p^{11} T^{2} \) |
| 73 | \( 1 - 1463791322 T + p^{11} T^{2} \) |
| 79 | \( 1 - 38116845680 T + p^{11} T^{2} \) |
| 83 | \( 1 - 29335099668 T + p^{11} T^{2} \) |
| 89 | \( 1 - 24992917110 T + p^{11} T^{2} \) |
| 97 | \( 1 - 75013568546 T + p^{11} T^{2} \) |
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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
−18.03776868120083516921539457264, −16.19726940025533186321036211697, −14.76364047014989640777640404045, −13.28469485632391117947647505139, −11.97926251213963967944860806496, −9.802567468383107143924341349161, −7.85370875205667359980381414999, −5.33651832334412859125614095350, −3.47688130619611814690570779390, 0,
3.47688130619611814690570779390, 5.33651832334412859125614095350, 7.85370875205667359980381414999, 9.802567468383107143924341349161, 11.97926251213963967944860806496, 13.28469485632391117947647505139, 14.76364047014989640777640404045, 16.19726940025533186321036211697, 18.03776868120083516921539457264
