| L(s) = 1 | − 24·2-s − 252·3-s − 1.47e3·4-s − 4.83e3·5-s + 6.04e3·6-s + 8.44e4·8-s − 1.13e5·9-s + 1.15e5·10-s + 5.34e5·11-s + 3.70e5·12-s + 5.77e5·13-s + 1.21e6·15-s + 9.87e5·16-s + 6.90e6·17-s + 2.72e6·18-s − 1.06e7·19-s + 7.10e6·20-s − 1.28e7·22-s + 1.86e7·23-s − 2.12e7·24-s − 2.54e7·25-s − 1.38e7·26-s + 7.32e7·27-s + 1.28e8·29-s − 2.92e7·30-s + 5.28e7·31-s − 1.96e8·32-s + ⋯ |
| L(s) = 1 | − 0.530·2-s − 0.598·3-s − 0.718·4-s − 0.691·5-s + 0.317·6-s + 0.911·8-s − 0.641·9-s + 0.366·10-s + 1.00·11-s + 0.430·12-s + 0.431·13-s + 0.413·15-s + 0.235·16-s + 1.17·17-s + 0.340·18-s − 0.987·19-s + 0.496·20-s − 0.530·22-s + 0.603·23-s − 0.545·24-s − 0.522·25-s − 0.228·26-s + 0.982·27-s + 1.16·29-s − 0.219·30-s + 0.331·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 3 p^{3} T + p^{11} T^{2} \) |
| 3 | \( 1 + 28 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 + 966 p T + p^{11} T^{2} \) |
| 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
−12.38816195505416672255210596968, −11.43327925700614111193786009449, −10.25071902320554841305195445199, −8.898287131797160073773009530050, −7.988437603884976742363203494243, −6.36407746423563155032025087677, −4.86958737255291716441841764611, −3.56196874919552369280423774233, −1.14797625714225285076913639649, 0,
1.14797625714225285076913639649, 3.56196874919552369280423774233, 4.86958737255291716441841764611, 6.36407746423563155032025087677, 7.988437603884976742363203494243, 8.898287131797160073773009530050, 10.25071902320554841305195445199, 11.43327925700614111193786009449, 12.38816195505416672255210596968
