| L(s) = 1 | − 9-s + 8·13-s + 14·17-s + 4·37-s + 10·41-s + 6·49-s + 12·53-s − 20·61-s + 18·73-s − 8·81-s − 10·89-s − 4·97-s + 4·101-s − 12·109-s + 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 14·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.21·13-s + 3.39·17-s + 0.657·37-s + 1.56·41-s + 6/7·49-s + 1.64·53-s − 2.56·61-s + 2.10·73-s − 8/9·81-s − 1.05·89-s − 0.406·97-s + 0.398·101-s − 1.14·109-s + 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.13·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.449777640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.449777640\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 129 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.474626111714987005105452428289, −9.367359995447237594093830192117, −8.752116593395323242801923147869, −8.351852504600398419862760046948, −7.996825105458390711585015727874, −7.77361375731057492250159914064, −7.29576770062729789200049563035, −6.81597589739661095394337101429, −6.13240985028103065463642227541, −5.97216720530962660687726711371, −5.47532704787015703891955995501, −5.44232202253705255954242354875, −4.52793571040102260089164909665, −3.95258882913647270386748568644, −3.73742767373973948123899984573, −3.04418283685822809452542412477, −2.92892832482561331793376992509, −1.90743131121157376759686576618, −1.07252425096184911019067718479, −0.971631826011971749588523265304,
0.971631826011971749588523265304, 1.07252425096184911019067718479, 1.90743131121157376759686576618, 2.92892832482561331793376992509, 3.04418283685822809452542412477, 3.73742767373973948123899984573, 3.95258882913647270386748568644, 4.52793571040102260089164909665, 5.44232202253705255954242354875, 5.47532704787015703891955995501, 5.97216720530962660687726711371, 6.13240985028103065463642227541, 6.81597589739661095394337101429, 7.29576770062729789200049563035, 7.77361375731057492250159914064, 7.996825105458390711585015727874, 8.351852504600398419862760046948, 8.752116593395323242801923147869, 9.367359995447237594093830192117, 9.474626111714987005105452428289
