| L(s) = 1 | + 4-s + 10·7-s + 12·13-s + 4·16-s − 5·25-s + 10·28-s − 6·31-s − 4·37-s − 4·43-s + 61·49-s + 12·52-s − 24·61-s + 11·64-s + 20·67-s − 4·79-s + 120·91-s + 48·97-s − 5·100-s − 6·103-s − 28·109-s + 40·112-s − 17·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.77·7-s + 3.32·13-s + 16-s − 25-s + 1.88·28-s − 1.07·31-s − 0.657·37-s − 0.609·43-s + 61/7·49-s + 1.66·52-s − 3.07·61-s + 11/8·64-s + 2.44·67-s − 0.450·79-s + 12.5·91-s + 4.87·97-s − 1/2·100-s − 0.591·103-s − 2.68·109-s + 3.77·112-s − 1.54·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.633736925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.633736925\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 17 T^{2} + 168 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 41 T^{2} + 1152 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 38 T^{2} + 603 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 67 T^{2} + 2808 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 34 T^{2} - 1053 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 58 T^{2} - 117 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 106 T^{2} + 4347 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.927709407128270379635016921442, −7.56600871008272304298893486481, −7.44118954379238510665287134487, −7.15588130562237895904015758190, −6.83567068908960919764116912653, −6.44596277746386511962424158404, −6.25469127971860898288268069103, −6.03731989250275003562667513781, −5.82069851885546312400547872077, −5.38210960989430636847805161569, −5.36808412690314150499155533024, −5.15448782288626654496235070064, −4.79147558419075862325611196995, −4.52983620796335113849128062076, −4.16721308387248423838542376528, −3.86496772589655833612195259618, −3.77305798286006032997579131066, −3.47323369269430713544560509064, −3.11389435560840924760071421220, −2.50991023953333009337863163064, −2.04054001506005170146877753108, −1.94106768991506357502833670448, −1.40442597066638605488782204109, −1.35352997901084349026376865349, −0.975915184238140800645085870741,
0.975915184238140800645085870741, 1.35352997901084349026376865349, 1.40442597066638605488782204109, 1.94106768991506357502833670448, 2.04054001506005170146877753108, 2.50991023953333009337863163064, 3.11389435560840924760071421220, 3.47323369269430713544560509064, 3.77305798286006032997579131066, 3.86496772589655833612195259618, 4.16721308387248423838542376528, 4.52983620796335113849128062076, 4.79147558419075862325611196995, 5.15448782288626654496235070064, 5.36808412690314150499155533024, 5.38210960989430636847805161569, 5.82069851885546312400547872077, 6.03731989250275003562667513781, 6.25469127971860898288268069103, 6.44596277746386511962424158404, 6.83567068908960919764116912653, 7.15588130562237895904015758190, 7.44118954379238510665287134487, 7.56600871008272304298893486481, 7.927709407128270379635016921442
