| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 35·16-s − 16·17-s − 4·19-s − 8·23-s − 16·31-s + 56·32-s − 64·34-s − 16·38-s − 32·46-s − 24·47-s − 12·49-s − 24·53-s − 8·61-s − 64·62-s + 84·64-s − 160·68-s − 40·76-s − 16·79-s − 24·83-s − 80·92-s − 96·94-s − 48·98-s − 96·106-s − 16·107-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 35/4·16-s − 3.88·17-s − 0.917·19-s − 1.66·23-s − 2.87·31-s + 9.89·32-s − 10.9·34-s − 2.59·38-s − 4.71·46-s − 3.50·47-s − 1.71·49-s − 3.29·53-s − 1.02·61-s − 8.12·62-s + 21/2·64-s − 19.4·68-s − 4.58·76-s − 1.80·79-s − 2.63·83-s − 8.34·92-s − 9.90·94-s − 4.84·98-s − 9.32·106-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 86 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 390 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 614 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4$ | \( 1 + 4 T^{2} + 486 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 4806 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 116 T^{2} + 6294 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 7110 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 4086 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 3510 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 28 T^{2} - 2010 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6246 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 340 T^{2} + 44694 T^{4} + 340 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 84 T^{2} - 586 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−5.80742519492580437630431964127, −5.60992904019846136249101856966, −5.26636700491076218577964732557, −5.26160731440865400173229999369, −5.03560580179963822372866474096, −4.75737678617815741084530795288, −4.70492343887779334889799690128, −4.59711051783282614574722854794, −4.41684708357054398286165760117, −4.10798333984428956213456727156, −4.03275512427787268705003381220, −3.83519589259441121093395807752, −3.81895567213535687052661989691, −3.33949799656411431151792646017, −3.26380560398458960770669074013, −3.15326982473643529882282292414, −2.80407351783986833524282687069, −2.58455952740621974854231633334, −2.42811607711859107251872654109, −2.21257853026140678874406563119, −2.03110620322102906332515267590, −1.73675145047941810413858599166, −1.52380750299865813559141647063, −1.44092654267693676921190026258, −1.36922478034992485458646426825, 0, 0, 0, 0,
1.36922478034992485458646426825, 1.44092654267693676921190026258, 1.52380750299865813559141647063, 1.73675145047941810413858599166, 2.03110620322102906332515267590, 2.21257853026140678874406563119, 2.42811607711859107251872654109, 2.58455952740621974854231633334, 2.80407351783986833524282687069, 3.15326982473643529882282292414, 3.26380560398458960770669074013, 3.33949799656411431151792646017, 3.81895567213535687052661989691, 3.83519589259441121093395807752, 4.03275512427787268705003381220, 4.10798333984428956213456727156, 4.41684708357054398286165760117, 4.59711051783282614574722854794, 4.70492343887779334889799690128, 4.75737678617815741084530795288, 5.03560580179963822372866474096, 5.26160731440865400173229999369, 5.26636700491076218577964732557, 5.60992904019846136249101856966, 5.80742519492580437630431964127
