| L(s) = 1 | − 2-s + 7-s + 9-s + 11-s − 14-s − 18-s − 22-s − 25-s + 32-s − 3·37-s − 2·43-s + 50-s − 3·53-s + 63-s − 64-s − 5·71-s + 3·74-s + 77-s + 3·79-s + 2·86-s + 99-s + 3·106-s − 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2-s + 7-s + 9-s + 11-s − 14-s − 18-s − 22-s − 25-s + 32-s − 3·37-s − 2·43-s + 50-s − 3·53-s + 63-s − 64-s − 5·71-s + 3·74-s + 77-s + 3·79-s + 2·86-s + 99-s + 3·106-s − 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2375373153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2375373153\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 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
−8.861998923407014778301819621520, −8.382716493259039290810826049305, −8.282448544448020597892557100282, −8.047660561423049747233429269554, −8.010151671126666524099950816493, −7.48737999795134532386452693412, −7.23162166458855112020158039630, −7.10056236599464798746278179550, −6.77362139046884596761242907804, −6.39784606343472149263810237620, −6.33127139745599844621435310411, −6.05805492837738807184103590942, −5.37295245900985956062009687080, −5.35187318976423925465995393069, −5.11172681900745300494131248828, −4.53028242276120442790334077655, −4.41063288806436562896926529562, −4.29444775627098395825160377739, −3.72973409947894333775293158240, −3.26488239840693834722000546922, −3.26219460831939368586986331495, −2.63008772983495359618069411294, −1.79160806752960980918348645419, −1.61543511764724728311692090592, −1.53128810455639629226018829549,
1.53128810455639629226018829549, 1.61543511764724728311692090592, 1.79160806752960980918348645419, 2.63008772983495359618069411294, 3.26219460831939368586986331495, 3.26488239840693834722000546922, 3.72973409947894333775293158240, 4.29444775627098395825160377739, 4.41063288806436562896926529562, 4.53028242276120442790334077655, 5.11172681900745300494131248828, 5.35187318976423925465995393069, 5.37295245900985956062009687080, 6.05805492837738807184103590942, 6.33127139745599844621435310411, 6.39784606343472149263810237620, 6.77362139046884596761242907804, 7.10056236599464798746278179550, 7.23162166458855112020158039630, 7.48737999795134532386452693412, 8.010151671126666524099950816493, 8.047660561423049747233429269554, 8.282448544448020597892557100282, 8.382716493259039290810826049305, 8.861998923407014778301819621520
