| L(s) = 1 | − 4·2-s + 4·4-s − 7·5-s + 16·8-s + 28·10-s − 25·11-s + 98·13-s − 64·16-s − 98·17-s − 119·19-s − 28·20-s + 100·22-s + 122·23-s + 214·25-s − 392·26-s − 146·29-s + 98·31-s + 64·32-s + 392·34-s − 289·37-s + 476·38-s − 112·40-s + 672·41-s + 614·43-s − 100·44-s − 488·46-s − 672·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.626·5-s + 0.707·8-s + 0.885·10-s − 0.685·11-s + 2.09·13-s − 16-s − 1.39·17-s − 1.43·19-s − 0.313·20-s + 0.969·22-s + 1.10·23-s + 1.71·25-s − 2.95·26-s − 0.934·29-s + 0.567·31-s + 0.353·32-s + 1.97·34-s − 1.28·37-s + 2.03·38-s − 0.442·40-s + 2.55·41-s + 2.17·43-s − 0.342·44-s − 1.56·46-s − 2.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1454892659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1454892659\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 7 T - 33 p T^{2} - 252 T^{3} + 24046 T^{4} - 252 p^{3} T^{5} - 33 p^{7} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 25 T + 171 T^{2} - 55200 T^{3} - 2397320 T^{4} - 55200 p^{3} T^{5} + 171 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 49 T + 3788 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 98 T + 2202 T^{2} - 237552 T^{3} - 16532417 T^{4} - 237552 p^{3} T^{5} + 2202 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 119 T - 2663 T^{2} + 369614 T^{3} + 138870796 T^{4} + 369614 p^{3} T^{5} - 2663 p^{6} T^{6} + 119 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 122 T - 3714 T^{2} + 699792 T^{3} + 16756087 T^{4} + 699792 p^{3} T^{5} - 3714 p^{6} T^{6} - 122 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 73 T + 47746 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 98 T - 14551 T^{2} + 3471846 T^{3} - 590152420 T^{4} + 3471846 p^{3} T^{5} - 14551 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 289 T - 17387 T^{2} - 115022 T^{3} + 3386108842 T^{4} - 115022 p^{3} T^{5} - 17387 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 672 T + 158834 T^{2} + 57189888 T^{3} + 28038541539 T^{4} + 57189888 p^{3} T^{5} + 158834 p^{6} T^{6} + 672 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 375 T - 781 T^{2} - 58630500 T^{3} - 23104532418 T^{4} - 58630500 p^{3} T^{5} - 781 p^{6} T^{6} + 375 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 763 T + 205407 T^{2} - 25938948 T^{3} - 19796216048 T^{4} - 25938948 p^{3} T^{5} + 205407 p^{6} T^{6} + 763 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 406 T - 274558 T^{2} - 5914608 T^{3} + 104132072759 T^{4} - 5914608 p^{3} T^{5} - 274558 p^{6} T^{6} + 406 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1041 T + 270341 T^{2} - 220498374 T^{3} + 245132424828 T^{4} - 220498374 p^{3} T^{5} + 270341 p^{6} T^{6} - 1041 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1652 T + 1360270 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 189 T - 297259 T^{2} - 84115206 T^{3} - 54354622146 T^{4} - 84115206 p^{3} T^{5} - 297259 p^{6} T^{6} + 189 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 524 T - 316753 T^{2} - 206848476 T^{3} - 28794145744 T^{4} - 206848476 p^{3} T^{5} - 316753 p^{6} T^{6} + 524 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 287 T + 781978 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2394 T + 3029186 T^{2} + 3093316128 T^{3} + 2763749199855 T^{4} + 3093316128 p^{3} T^{5} + 3029186 p^{6} T^{6} + 2394 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 63 T + 667132 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 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
−6.93371180572883851079238865532, −6.68879722135631091313959214588, −6.58913927395881682346007454728, −6.13009673672508674707719343371, −6.02357290737109661431326319523, −5.88367144442355866166647187246, −5.64555839740390114981233787666, −5.05820013297250289599540558898, −4.93459839390444368859145007906, −4.66867651908309278526935639981, −4.65284490165533835116164988548, −4.13110711761321918622436693131, −3.92997883640711184495508414464, −3.72986433601185443951104677809, −3.70806217806283388848049930181, −2.91672180158096617827312958212, −2.66977329168281945506787728865, −2.61291499748378412180360482372, −2.54606952673195964147098399612, −1.55599650052979551854300108149, −1.44464485161264357297756313280, −1.28160882334711784208958997830, −1.14831987697699292890258770638, −0.23191399094423486143631332551, −0.16799044867191723854931161457,
0.16799044867191723854931161457, 0.23191399094423486143631332551, 1.14831987697699292890258770638, 1.28160882334711784208958997830, 1.44464485161264357297756313280, 1.55599650052979551854300108149, 2.54606952673195964147098399612, 2.61291499748378412180360482372, 2.66977329168281945506787728865, 2.91672180158096617827312958212, 3.70806217806283388848049930181, 3.72986433601185443951104677809, 3.92997883640711184495508414464, 4.13110711761321918622436693131, 4.65284490165533835116164988548, 4.66867651908309278526935639981, 4.93459839390444368859145007906, 5.05820013297250289599540558898, 5.64555839740390114981233787666, 5.88367144442355866166647187246, 6.02357290737109661431326319523, 6.13009673672508674707719343371, 6.58913927395881682346007454728, 6.68879722135631091313959214588, 6.93371180572883851079238865532
