| L(s) = 1 | − 4·5-s + 9-s − 4·11-s + 8·19-s + 2·25-s + 12·37-s − 8·43-s − 4·45-s − 14·49-s − 4·53-s + 16·55-s + 16·79-s + 81-s + 8·83-s − 12·89-s − 32·95-s + 4·97-s − 4·99-s + 24·107-s + 36·113-s + 5·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s − 1.20·11-s + 1.83·19-s + 2/5·25-s + 1.97·37-s − 1.21·43-s − 0.596·45-s − 2·49-s − 0.549·53-s + 2.15·55-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 1.27·89-s − 3.28·95-s + 0.406·97-s − 0.402·99-s + 2.32·107-s + 3.38·113-s + 5/11·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8568211475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8568211475\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.646412525003839411544338452938, −8.632814053171275790838666005226, −8.173282349834764667604075089133, −7.88365108150257423936793116264, −7.41145043458246524552596785021, −7.26958585488936859387169650645, −6.34034973016640940830304663850, −5.85561327512449935358731086305, −5.01357758335939619070346679818, −4.76750339316057597575339371032, −4.08582339913610742428343396917, −3.30728356810877078064881030636, −3.14826068230388029805668446658, −1.97854038896010733538639767531, −0.63207175719826203780734613127,
0.63207175719826203780734613127, 1.97854038896010733538639767531, 3.14826068230388029805668446658, 3.30728356810877078064881030636, 4.08582339913610742428343396917, 4.76750339316057597575339371032, 5.01357758335939619070346679818, 5.85561327512449935358731086305, 6.34034973016640940830304663850, 7.26958585488936859387169650645, 7.41145043458246524552596785021, 7.88365108150257423936793116264, 8.173282349834764667604075089133, 8.632814053171275790838666005226, 9.646412525003839411544338452938
