| L(s) = 1 | + 2·3-s + 9-s + 4·13-s − 12·23-s + 25-s − 4·27-s + 4·37-s + 8·39-s + 12·47-s − 10·49-s − 24·59-s + 4·61-s − 24·69-s + 24·71-s + 4·73-s + 2·75-s − 11·81-s − 12·83-s + 4·97-s + 12·107-s + 4·109-s + 8·111-s + 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.10·13-s − 2.50·23-s + 1/5·25-s − 0.769·27-s + 0.657·37-s + 1.28·39-s + 1.75·47-s − 1.42·49-s − 3.12·59-s + 0.512·61-s − 2.88·69-s + 2.84·71-s + 0.468·73-s + 0.230·75-s − 1.22·81-s − 1.31·83-s + 0.406·97-s + 1.16·107-s + 0.383·109-s + 0.759·111-s + 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492977957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492977957\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−11.10973086408237495845181170606, −10.66869217043212474435745329357, −9.749524176059453166876684221310, −9.683565298540966130897522415203, −8.740293324354028093320034492248, −8.536040373596851277524077678909, −7.71273110823499542846308181544, −7.64589778281564359800589548721, −6.27087624192875571051851265258, −6.21444115782927123192608967686, −5.19647848534431193907436500640, −4.12250433368686324236236368171, −3.71119793411816721603957340078, −2.76929890617261215013507568311, −1.81793015252092636076156145980,
1.81793015252092636076156145980, 2.76929890617261215013507568311, 3.71119793411816721603957340078, 4.12250433368686324236236368171, 5.19647848534431193907436500640, 6.21444115782927123192608967686, 6.27087624192875571051851265258, 7.64589778281564359800589548721, 7.71273110823499542846308181544, 8.536040373596851277524077678909, 8.740293324354028093320034492248, 9.683565298540966130897522415203, 9.749524176059453166876684221310, 10.66869217043212474435745329357, 11.10973086408237495845181170606
