| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s − 6·9-s + 2·10-s − 11-s − 16-s − 6·18-s − 8·19-s − 2·20-s − 22-s + 3·25-s + 5·32-s + 6·36-s − 4·37-s − 8·38-s − 6·40-s + 8·43-s + 44-s − 12·45-s − 14·49-s + 3·50-s − 4·53-s − 2·55-s + 7·64-s + 18·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s − 2·9-s + 0.632·10-s − 0.301·11-s − 1/4·16-s − 1.41·18-s − 1.83·19-s − 0.447·20-s − 0.213·22-s + 3/5·25-s + 0.883·32-s + 36-s − 0.657·37-s − 1.29·38-s − 0.948·40-s + 1.21·43-s + 0.150·44-s − 1.78·45-s − 2·49-s + 0.424·50-s − 0.549·53-s − 0.269·55-s + 7/8·64-s + 2.12·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276190995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276190995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.510783714703124870590796818437, −8.248642768475844318802620430882, −7.69556880125895993906919048601, −6.92114469673781934514976443303, −6.21371933475241794414167140387, −6.16821463973303244938343944630, −5.79962506110486358982246461760, −5.09766059609940450216711100862, −4.87966949514469953615819754787, −4.30523069678046427722855688335, −3.37416713721777180685271968198, −3.25423179226253980082861279896, −2.37289889085752151543981096662, −2.04716871831388657730114879319, −0.49631415514383905112999669566,
0.49631415514383905112999669566, 2.04716871831388657730114879319, 2.37289889085752151543981096662, 3.25423179226253980082861279896, 3.37416713721777180685271968198, 4.30523069678046427722855688335, 4.87966949514469953615819754787, 5.09766059609940450216711100862, 5.79962506110486358982246461760, 6.16821463973303244938343944630, 6.21371933475241794414167140387, 6.92114469673781934514976443303, 7.69556880125895993906919048601, 8.248642768475844318802620430882, 8.510783714703124870590796818437
