| L(s) = 1 | − 243·3-s + 1.19e3·5-s + 1.84e4·7-s + 5.90e4·9-s + 1.35e5·11-s − 8.48e5·13-s − 2.89e5·15-s − 7.12e6·17-s − 5.04e6·19-s − 4.49e6·21-s − 1.48e7·23-s − 4.74e7·25-s − 1.43e7·27-s − 1.15e8·29-s − 1.63e8·31-s − 3.30e7·33-s + 2.19e7·35-s − 2.23e8·37-s + 2.06e8·39-s + 1.05e8·41-s + 1.41e9·43-s + 7.02e7·45-s + 2.46e9·47-s − 1.63e9·49-s + 1.73e9·51-s − 4.83e8·53-s + 1.61e8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.170·5-s + 0.415·7-s + 1/3·9-s + 0.254·11-s − 0.633·13-s − 0.0983·15-s − 1.21·17-s − 0.467·19-s − 0.239·21-s − 0.482·23-s − 0.970·25-s − 0.192·27-s − 1.04·29-s − 1.02·31-s − 0.146·33-s + 0.0707·35-s − 0.530·37-s + 0.365·39-s + 0.142·41-s + 1.47·43-s + 0.0567·45-s + 1.57·47-s − 0.827·49-s + 0.702·51-s − 0.158·53-s + 0.0433·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + p^{5} T \) |
| good | 5 | \( 1 - 238 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 2640 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 135884 T + p^{11} T^{2} \) |
| 13 | \( 1 + 848186 T + p^{11} T^{2} \) |
| 17 | \( 1 + 7124606 T + p^{11} T^{2} \) |
| 19 | \( 1 + 5046316 T + p^{11} T^{2} \) |
| 23 | \( 1 + 14891224 T + p^{11} T^{2} \) |
| 29 | \( 1 + 115001346 T + p^{11} T^{2} \) |
| 31 | \( 1 + 163990552 T + p^{11} T^{2} \) |
| 37 | \( 1 + 223622178 T + p^{11} T^{2} \) |
| 41 | \( 1 - 105358314 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1419475852 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2469276960 T + p^{11} T^{2} \) |
| 53 | \( 1 + 483704986 T + p^{11} T^{2} \) |
| 59 | \( 1 - 6151842476 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7532732282 T + p^{11} T^{2} \) |
| 67 | \( 1 + 8764949068 T + p^{11} T^{2} \) |
| 71 | \( 1 + 10401627752 T + p^{11} T^{2} \) |
| 73 | \( 1 + 31738391270 T + p^{11} T^{2} \) |
| 79 | \( 1 + 39880016072 T + p^{11} T^{2} \) |
| 83 | \( 1 - 13513323988 T + p^{11} T^{2} \) |
| 89 | \( 1 - 81514517226 T + p^{11} T^{2} \) |
| 97 | \( 1 - 30783027074 T + p^{11} T^{2} \) |
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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
−14.52873631700986125295345492010, −13.10942784114493102816329186827, −11.79316047981962020936858064509, −10.64385650997440387687936085212, −9.132837553853655715396314685310, −7.37458332416202712812767250460, −5.83384026818917246306933764205, −4.28981422286922868645727314872, −1.96382331205627020455408224380, 0,
1.96382331205627020455408224380, 4.28981422286922868645727314872, 5.83384026818917246306933764205, 7.37458332416202712812767250460, 9.132837553853655715396314685310, 10.64385650997440387687936085212, 11.79316047981962020936858064509, 13.10942784114493102816329186827, 14.52873631700986125295345492010
