| L(s) = 1 | − 243·3-s + 3.63e3·5-s − 3.29e4·7-s + 5.90e4·9-s + 7.58e5·11-s − 2.48e6·13-s − 8.82e5·15-s + 8.29e6·17-s + 1.08e7·19-s + 8.00e6·21-s − 2.05e7·23-s − 3.56e7·25-s − 1.43e7·27-s + 2.88e7·29-s − 1.50e8·31-s − 1.84e8·33-s − 1.19e8·35-s − 3.19e8·37-s + 6.03e8·39-s − 3.68e8·41-s − 6.20e8·43-s + 2.14e8·45-s − 2.76e9·47-s − 8.92e8·49-s − 2.01e9·51-s − 2.68e8·53-s + 2.75e9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.519·5-s − 0.740·7-s + 1/3·9-s + 1.42·11-s − 1.85·13-s − 0.299·15-s + 1.41·17-s + 1.00·19-s + 0.427·21-s − 0.665·23-s − 0.730·25-s − 0.192·27-s + 0.260·29-s − 0.944·31-s − 0.820·33-s − 0.384·35-s − 0.758·37-s + 1.07·39-s − 0.496·41-s − 0.643·43-s + 0.173·45-s − 1.75·47-s − 0.451·49-s − 0.817·51-s − 0.0881·53-s + 0.737·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 726 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 32936 T + p^{11} T^{2} \) |
| 11 | \( 1 - 758748 T + p^{11} T^{2} \) |
| 13 | \( 1 + 2482858 T + p^{11} T^{2} \) |
| 17 | \( 1 - 8290386 T + p^{11} T^{2} \) |
| 19 | \( 1 - 10867300 T + p^{11} T^{2} \) |
| 23 | \( 1 + 20539272 T + p^{11} T^{2} \) |
| 29 | \( 1 - 28814550 T + p^{11} T^{2} \) |
| 31 | \( 1 + 150501392 T + p^{11} T^{2} \) |
| 37 | \( 1 + 8645722 p T + p^{11} T^{2} \) |
| 41 | \( 1 + 368008998 T + p^{11} T^{2} \) |
| 43 | \( 1 + 620469572 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2763110256 T + p^{11} T^{2} \) |
| 53 | \( 1 + 268284258 T + p^{11} T^{2} \) |
| 59 | \( 1 + 1672894740 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7787197498 T + p^{11} T^{2} \) |
| 67 | \( 1 + 18706694156 T + p^{11} T^{2} \) |
| 71 | \( 1 - 8346990888 T + p^{11} T^{2} \) |
| 73 | \( 1 - 19641746522 T + p^{11} T^{2} \) |
| 79 | \( 1 - 5873807200 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8492558172 T + p^{11} T^{2} \) |
| 89 | \( 1 - 75527864010 T + p^{11} T^{2} \) |
| 97 | \( 1 + 82356782494 T + p^{11} T^{2} \) |
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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
−12.39263412549035734994621022648, −11.80341755242482456248162475670, −9.975662948204413836263602815556, −9.527999739638919330730814025303, −7.48000078596117777440198416397, −6.31257221177734533056402236106, −5.10787002745666269545215595830, −3.40274001862606195843693918630, −1.59540408093074536653483453947, 0,
1.59540408093074536653483453947, 3.40274001862606195843693918630, 5.10787002745666269545215595830, 6.31257221177734533056402236106, 7.48000078596117777440198416397, 9.527999739638919330730814025303, 9.975662948204413836263602815556, 11.80341755242482456248162475670, 12.39263412549035734994621022648
