| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s + 8·9-s + 35·16-s − 32·18-s + 16·23-s + 8·25-s − 32·29-s − 56·32-s + 80·36-s − 64·46-s − 14·49-s − 32·50-s + 128·58-s + 84·64-s − 32·71-s − 160·72-s + 30·81-s + 160·92-s + 56·98-s + 80·100-s − 320·116-s + 44·121-s + 127-s − 120·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s + 8/3·9-s + 35/4·16-s − 7.54·18-s + 3.33·23-s + 8/5·25-s − 5.94·29-s − 9.89·32-s + 40/3·36-s − 9.43·46-s − 2·49-s − 4.52·50-s + 16.8·58-s + 21/2·64-s − 3.79·71-s − 18.8·72-s + 10/3·81-s + 16.6·92-s + 5.65·98-s + 8·100-s − 29.7·116-s + 4·121-s + 0.0887·127-s − 10.6·128-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5934152104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5934152104\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 180 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.595898263026719871588997885137, −8.356604533998303811219512243228, −7.86715343361403798835407394038, −7.55569508793597176489721116091, −7.35889979155232871861819320709, −7.27445595820734166925216896060, −7.26668318379518254919782398522, −6.87323420803761577597016644613, −6.83759597024854198131915824309, −6.31397406787443349020787249560, −5.98996458540410725304662528252, −5.73449253747493668126872226590, −5.34744499399178395964628896284, −5.11190248178823588154160193147, −4.64460010408824779749993389760, −4.38188763453027629351225481002, −4.03000668436828988501673781916, −3.37895924201454595072450050047, −3.22964408242373636615264868968, −3.10003972301179746458859010178, −2.26255319679862247724222898727, −1.85258843189487934132622444710, −1.53845996156559291627729904806, −1.38766537805611112302199673759, −0.58993131989671849072367259498,
0.58993131989671849072367259498, 1.38766537805611112302199673759, 1.53845996156559291627729904806, 1.85258843189487934132622444710, 2.26255319679862247724222898727, 3.10003972301179746458859010178, 3.22964408242373636615264868968, 3.37895924201454595072450050047, 4.03000668436828988501673781916, 4.38188763453027629351225481002, 4.64460010408824779749993389760, 5.11190248178823588154160193147, 5.34744499399178395964628896284, 5.73449253747493668126872226590, 5.98996458540410725304662528252, 6.31397406787443349020787249560, 6.83759597024854198131915824309, 6.87323420803761577597016644613, 7.26668318379518254919782398522, 7.27445595820734166925216896060, 7.35889979155232871861819320709, 7.55569508793597176489721116091, 7.86715343361403798835407394038, 8.356604533998303811219512243228, 8.595898263026719871588997885137
