| L(s) = 1 | − 2·7-s + 4·16-s + 2·19-s + 8·31-s + 20·37-s − 11·49-s + 32·67-s + 34·73-s + 28·97-s + 14·103-s + 4·109-s − 8·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 16-s + 0.458·19-s + 1.43·31-s + 3.28·37-s − 1.57·49-s + 3.90·67-s + 3.97·73-s + 2.84·97-s + 1.37·103-s + 0.383·109-s − 0.755·112-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-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 − 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.525257290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.525257290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60053272382129399922093824364, −7.02462047976336185615941628713, −6.83826496811233019553964824341, −6.48000644834023970202986002486, −6.46419370363395651541708712240, −6.29517896123085508196286536881, −6.05197679997966541460323903949, −5.83739251376929596833582030266, −5.42920551344811077508652339575, −5.18963736543501750401590533506, −5.00726830393899970755538021258, −4.74858355937925129214639943774, −4.63940697749863991189258214257, −4.05353285820419653327569098455, −3.93429133252179873612956222403, −3.62492036696822417255868747479, −3.46254379533119450246343269706, −3.14840868902032201074568574676, −2.77463434166321514534145504697, −2.46101868647632704624854464502, −2.29616486311959245667257003422, −1.89508523515430088313551252179, −1.23820245675931959769306271427, −0.818887179922443872180845300868, −0.66447354808489655831845709755,
0.66447354808489655831845709755, 0.818887179922443872180845300868, 1.23820245675931959769306271427, 1.89508523515430088313551252179, 2.29616486311959245667257003422, 2.46101868647632704624854464502, 2.77463434166321514534145504697, 3.14840868902032201074568574676, 3.46254379533119450246343269706, 3.62492036696822417255868747479, 3.93429133252179873612956222403, 4.05353285820419653327569098455, 4.63940697749863991189258214257, 4.74858355937925129214639943774, 5.00726830393899970755538021258, 5.18963736543501750401590533506, 5.42920551344811077508652339575, 5.83739251376929596833582030266, 6.05197679997966541460323903949, 6.29517896123085508196286536881, 6.46419370363395651541708712240, 6.48000644834023970202986002486, 6.83826496811233019553964824341, 7.02462047976336185615941628713, 7.60053272382129399922093824364
