| L(s) = 1 | − 4·9-s + 2·23-s − 10·25-s + 4·29-s + 12·37-s − 16·43-s − 4·53-s + 16·67-s + 16·79-s + 7·81-s + 28·109-s + 4·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 0.417·23-s − 2·25-s + 0.742·29-s + 1.97·37-s − 2.43·43-s − 0.549·53-s + 1.95·67-s + 1.80·79-s + 7/9·81-s + 2.68·109-s + 0.376·113-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040230114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040230114\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−7.945769079337080906020480981471, −7.78930687793646530915720468390, −7.14314535589195570401368519809, −6.90608195789212644761787726064, −6.34341111110808543273734602883, −6.32462479244687803064474945142, −5.78553991052182409641974429607, −5.65478780180231530985192297060, −5.03184257023399371026197094672, −4.93982838751652403693876594586, −4.45556856708408340307816115769, −3.99009473640858634473736416749, −3.47906642763832717855489659386, −3.42234916764982104307155337370, −2.64784146429291841833360449760, −2.64051866298409372288134129927, −1.90512458480281273839524024405, −1.68181404569708699150966918492, −0.76555528452522427748955171574, −0.42616788008840568177434102071,
0.42616788008840568177434102071, 0.76555528452522427748955171574, 1.68181404569708699150966918492, 1.90512458480281273839524024405, 2.64051866298409372288134129927, 2.64784146429291841833360449760, 3.42234916764982104307155337370, 3.47906642763832717855489659386, 3.99009473640858634473736416749, 4.45556856708408340307816115769, 4.93982838751652403693876594586, 5.03184257023399371026197094672, 5.65478780180231530985192297060, 5.78553991052182409641974429607, 6.32462479244687803064474945142, 6.34341111110808543273734602883, 6.90608195789212644761787726064, 7.14314535589195570401368519809, 7.78930687793646530915720468390, 7.945769079337080906020480981471
