| L(s) = 1 | + 2·2-s + 4-s + 2·5-s − 2·8-s − 9-s + 4·10-s − 2·11-s − 20·13-s − 4·16-s + 12·17-s − 2·18-s − 6·19-s + 2·20-s − 4·22-s + 2·23-s + 4·25-s − 40·26-s + 4·29-s − 8·31-s − 2·32-s + 24·34-s − 36-s − 2·37-s − 12·38-s − 4·40-s − 12·41-s − 16·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.707·8-s − 1/3·9-s + 1.26·10-s − 0.603·11-s − 5.54·13-s − 16-s + 2.91·17-s − 0.471·18-s − 1.37·19-s + 0.447·20-s − 0.852·22-s + 0.417·23-s + 4/5·25-s − 7.84·26-s + 0.742·29-s − 1.43·31-s − 0.353·32-s + 4.11·34-s − 1/6·36-s − 0.328·37-s − 1.94·38-s − 0.632·40-s − 1.87·41-s − 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.586561075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586561075\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T + 12 T^{3} - 29 T^{4} + 12 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} + 12 T^{3} + 555 T^{4} + 12 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 12 p T^{5} - 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 126 T^{2} - 576 T^{3} + 2659 T^{4} - 576 p T^{5} + 126 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 99 T^{2} + 12 T^{3} + 8800 T^{4} + 12 p T^{5} - 99 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 18 T + 149 T^{2} - 954 T^{3} + 7140 T^{4} - 954 p T^{5} + 149 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 376 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 14 T + 184 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 6 T - 112 T^{2} - 12 T^{3} + 13947 T^{4} - 12 p T^{5} - 112 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 18 T^{2} - 768 T^{3} - 6893 T^{4} - 768 p T^{5} - 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 22 T + 287 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−7.04910239769335878197846050192, −6.75033189537099838005175777764, −6.71792122841608023526109343554, −6.58242604232343421208685875340, −5.98579406441658306455498280898, −5.56113314390196854809643239412, −5.52755961596187646416650777992, −5.36247701671764863240333130054, −5.34548159213326751067708753445, −5.18964317795485149413918683492, −4.66349942605139029108726604584, −4.65866126425849506085650753326, −4.62640196067203972810318152721, −4.13160352832875838009458824571, −3.69251203511804993708459772335, −3.42866928781008433003896182421, −3.23800913150907555667524250169, −3.09421587999297920405235900063, −2.56675654127348811436120391462, −2.35385933149064249157588575275, −2.24997757033258156038998580017, −2.07420813306571977359161571275, −1.55907869039881875742510740922, −0.58201281571905823276832992890, −0.48118700019040031325469807170,
0.48118700019040031325469807170, 0.58201281571905823276832992890, 1.55907869039881875742510740922, 2.07420813306571977359161571275, 2.24997757033258156038998580017, 2.35385933149064249157588575275, 2.56675654127348811436120391462, 3.09421587999297920405235900063, 3.23800913150907555667524250169, 3.42866928781008433003896182421, 3.69251203511804993708459772335, 4.13160352832875838009458824571, 4.62640196067203972810318152721, 4.65866126425849506085650753326, 4.66349942605139029108726604584, 5.18964317795485149413918683492, 5.34548159213326751067708753445, 5.36247701671764863240333130054, 5.52755961596187646416650777992, 5.56113314390196854809643239412, 5.98579406441658306455498280898, 6.58242604232343421208685875340, 6.71792122841608023526109343554, 6.75033189537099838005175777764, 7.04910239769335878197846050192
