| L(s) = 1 | − 3·2-s − 2·3-s − 2·4-s + 6·6-s + 24·7-s + 15·8-s + 27·9-s − 24·11-s + 4·12-s − 72·14-s − 15·16-s − 117·17-s − 81·18-s + 198·19-s − 48·21-s + 72·22-s + 78·23-s − 30·24-s + 247·25-s − 154·27-s − 48·28-s + 141·29-s + 120·32-s + 48·33-s + 351·34-s − 54·36-s + 249·37-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.384·3-s − 1/4·4-s + 0.408·6-s + 1.29·7-s + 0.662·8-s + 9-s − 0.657·11-s + 0.0962·12-s − 1.37·14-s − 0.234·16-s − 1.66·17-s − 1.06·18-s + 2.39·19-s − 0.498·21-s + 0.697·22-s + 0.707·23-s − 0.255·24-s + 1.97·25-s − 1.09·27-s − 0.323·28-s + 0.902·29-s + 0.662·32-s + 0.253·33-s + 1.77·34-s − 1/4·36-s + 1.10·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.106921256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106921256\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 11 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 247 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 535 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 24 T + 1523 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 117 T + 8776 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 198 T + 19927 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 141 T - 4508 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 249 T + 71320 T^{2} - 249 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 471 T + 142868 T^{2} + 471 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 116818 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 492 T + 286067 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1362 T + 919111 T^{2} - 1362 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1830 T + 1474211 T^{2} + 1830 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 567359 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 519766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1692 T + 1659257 T^{2} - 1692 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 348 T + 953041 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.56419516355156520454756162022, −11.71944963089547327153897892355, −11.59275452915860328983811817983, −11.01728697458626857279640365103, −10.39028590236463935327681274345, −10.16052625422979483368065081463, −9.274941552613008166721962211635, −9.240192942577938792149626266221, −8.415194175077872610888586081850, −8.136885324681471532042815750738, −7.45683561700817193666835789616, −6.99539880623038828219996358249, −6.40917635934714065641302057123, −5.24446126832002427994338826621, −4.88911036549834203794276131516, −4.62366077445546092077662277598, −3.48454413231220453678319345641, −2.43282159643065322809981699102, −1.31414323649380026611142289833, −0.69775681349343177399839854730,
0.69775681349343177399839854730, 1.31414323649380026611142289833, 2.43282159643065322809981699102, 3.48454413231220453678319345641, 4.62366077445546092077662277598, 4.88911036549834203794276131516, 5.24446126832002427994338826621, 6.40917635934714065641302057123, 6.99539880623038828219996358249, 7.45683561700817193666835789616, 8.136885324681471532042815750738, 8.415194175077872610888586081850, 9.240192942577938792149626266221, 9.274941552613008166721962211635, 10.16052625422979483368065081463, 10.39028590236463935327681274345, 11.01728697458626857279640365103, 11.59275452915860328983811817983, 11.71944963089547327153897892355, 12.56419516355156520454756162022
