| L(s) = 1 | − 4·2-s + 4·4-s + 10·5-s − 12·7-s + 16·8-s − 18·9-s − 40·10-s − 18·11-s − 12·13-s + 48·14-s − 64·16-s + 72·18-s − 2·19-s + 40·20-s + 72·22-s + 26·23-s + 25·25-s + 48·26-s − 48·28-s + 64·32-s − 120·35-s − 72·36-s + 54·37-s + 8·38-s + 160·40-s + 156·41-s − 72·44-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·5-s − 1.71·7-s + 2·8-s − 2·9-s − 4·10-s − 1.63·11-s − 0.923·13-s + 24/7·14-s − 4·16-s + 4·18-s − 0.105·19-s + 2·20-s + 3.27·22-s + 1.13·23-s + 25-s + 1.84·26-s − 1.71·28-s + 2·32-s − 3.42·35-s − 2·36-s + 1.45·37-s + 4/19·38-s + 4·40-s + 3.80·41-s − 1.63·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.002960379484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002960379484\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2}( 1 - 18 T + 203 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T - 133 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 - 2 T - 357 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2}( 1 + 26 T + 147 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2}( 1 + 54 T + 1547 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T + 4403 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2}( 1 - 86 T + 5187 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2}( 1 - 74 T + 2667 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 78 T + 2603 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T - 7597 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−8.555077674432452507116617186627, −8.083039078195432541908009190219, −8.065266670764993334150853980615, −7.76120813341322811142949529243, −7.57321503586881186957937393097, −7.15741331940307878325916978711, −7.05670552500071301107168977819, −6.39163756237134769078632172332, −6.25230998121198836741627377587, −6.21521849849379008161196646274, −5.69150843036458160746717959414, −5.64507185635192368636450633179, −5.24761550028313021063266182691, −4.91554374977072707936655969686, −4.70509312683717951911123589070, −4.31096018932013282170651712033, −3.89279239988464821541101949021, −3.08401934029036555265804750858, −2.97378509095550718996165204813, −2.73314686018658411575453483298, −2.42211465420193280596640743267, −1.92233668208660893604018172173, −1.44236713894353086726431621597, −0.70573575296470608051574651648, −0.02735027823089611273577624605,
0.02735027823089611273577624605, 0.70573575296470608051574651648, 1.44236713894353086726431621597, 1.92233668208660893604018172173, 2.42211465420193280596640743267, 2.73314686018658411575453483298, 2.97378509095550718996165204813, 3.08401934029036555265804750858, 3.89279239988464821541101949021, 4.31096018932013282170651712033, 4.70509312683717951911123589070, 4.91554374977072707936655969686, 5.24761550028313021063266182691, 5.64507185635192368636450633179, 5.69150843036458160746717959414, 6.21521849849379008161196646274, 6.25230998121198836741627377587, 6.39163756237134769078632172332, 7.05670552500071301107168977819, 7.15741331940307878325916978711, 7.57321503586881186957937393097, 7.76120813341322811142949529243, 8.065266670764993334150853980615, 8.083039078195432541908009190219, 8.555077674432452507116617186627
