| L(s) = 1 | + 18·3-s + 243·9-s + 284·11-s − 866·25-s + 2.91e3·27-s + 5.11e3·33-s + 4.79e3·49-s − 1.37e4·59-s + 1.63e4·73-s − 1.55e4·75-s + 3.28e4·81-s − 8.35e3·83-s + 3.45e4·97-s + 6.90e4·99-s − 3.10e4·107-s + 3.12e4·121-s + 127-s + 131-s + 137-s + 139-s + 8.63e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.71e4·169-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2.34·11-s − 1.38·25-s + 4·27-s + 4.69·33-s + 1.99·49-s − 3.94·59-s + 3.06·73-s − 2.77·75-s + 5·81-s − 1.21·83-s + 3.67·97-s + 7.04·99-s − 2.70·107-s + 2.13·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.99·147-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(9.592886902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.592886902\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 866 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4798 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 142 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 745438 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1618558 T^{2} + p^{8} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5425438 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6862 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8158 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 5237762 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4178 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17282 T + p^{4} T^{2} )^{2} \) |
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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
−10.77421225751000645171092822645, −10.39220456587035973264790395994, −9.675016392611588363553641865439, −9.480336648589461586716515012997, −8.979456944097746033435637172199, −8.954051723152873209420788442368, −8.017948514145468185536971421514, −7.954388572052167239153067471484, −7.23983670503207734541515943751, −6.85568308932687972245162355129, −6.33871604860404353897731076579, −5.81072046107287518453662502688, −4.69811092574245367116374689366, −4.33774680984265577160735086745, −3.68466307840023123621726877939, −3.54750103536048768627815563457, −2.70645891677297904096548516511, −1.94611668112733102951553066352, −1.54562491351077945590712554728, −0.808910801807589033063655189094,
0.808910801807589033063655189094, 1.54562491351077945590712554728, 1.94611668112733102951553066352, 2.70645891677297904096548516511, 3.54750103536048768627815563457, 3.68466307840023123621726877939, 4.33774680984265577160735086745, 4.69811092574245367116374689366, 5.81072046107287518453662502688, 6.33871604860404353897731076579, 6.85568308932687972245162355129, 7.23983670503207734541515943751, 7.954388572052167239153067471484, 8.017948514145468185536971421514, 8.954051723152873209420788442368, 8.979456944097746033435637172199, 9.480336648589461586716515012997, 9.675016392611588363553641865439, 10.39220456587035973264790395994, 10.77421225751000645171092822645
