| L(s) = 1 | + 3·2-s + 3-s + 7·4-s + 3·6-s − 4·7-s + 15·8-s − 2·11-s + 7·12-s + 9·13-s − 12·14-s + 30·16-s + 13·17-s − 4·21-s − 6·22-s − 23-s + 15·24-s + 10·25-s + 27·26-s − 28·28-s − 15·29-s − 7·31-s + 57·32-s − 2·33-s + 39·34-s + 3·37-s + 9·39-s − 12·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 7/2·4-s + 1.22·6-s − 1.51·7-s + 5.30·8-s − 0.603·11-s + 2.02·12-s + 2.49·13-s − 3.20·14-s + 15/2·16-s + 3.15·17-s − 0.872·21-s − 1.27·22-s − 0.208·23-s + 3.06·24-s + 2·25-s + 5.29·26-s − 5.29·28-s − 2.78·29-s − 1.25·31-s + 10.0·32-s − 0.348·33-s + 6.68·34-s + 0.493·37-s + 1.44·39-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.50759480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.50759480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 9 T + 48 T^{2} - 235 T^{3} + 1011 T^{4} - 235 p T^{5} + 48 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 13 T + 52 T^{2} + 25 T^{3} - 729 T^{4} + 25 p T^{5} + 52 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 70 p T^{5} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + T - 17 T^{2} + 65 T^{3} + 576 T^{4} + 65 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + 15 T + 106 T^{2} + 675 T^{3} + 4171 T^{4} + 675 p T^{5} + 106 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 119 p T^{5} + 3 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 3 T + 17 T^{2} - 225 T^{3} + 2116 T^{4} - 225 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 444 p T^{5} + 53 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 160 p T^{5} + 43 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 530 p T^{5} + 131 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 8 T + 123 T^{2} - 766 T^{3} + 10325 T^{4} - 766 p T^{5} + 123 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 895 p T^{5} + 12 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 349 p T^{5} - 2 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 21 T + 93 T^{2} - 1405 T^{3} - 22044 T^{4} - 1405 p T^{5} + 93 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 1505 p T^{5} + 231 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 16 T + 173 T^{2} + 2200 T^{3} + 26921 T^{4} + 2200 p T^{5} + 173 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 200 p T^{5} - 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 865 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.84431998514043485500010396307, −7.38469421984987685418205477244, −7.28880710560473519556560070841, −6.91771776750852167465237574510, −6.84250419727852991433220720269, −6.82178126111040023748214800129, −6.02197424421362615696617875634, −5.99838225634515335205206757929, −5.91417708724242520864586791079, −5.62259359932216222501328682576, −5.51621562978604837944520528828, −5.12486970892762200839904860534, −5.00741716684383703557451421431, −4.37422959629813724442148635519, −4.05728632087198794572034964292, −3.97734129503180064768784414404, −3.66494612750793932267745417923, −3.22043251743417688051139364853, −3.12673878712068688196868780349, −3.03246709456858498475918707015, −2.95321747653964698216294685346, −1.93213126305064007059107443207, −1.85836653948862329741753387911, −1.36045817475820196998737946707, −0.984431225127897406254179517371,
0.984431225127897406254179517371, 1.36045817475820196998737946707, 1.85836653948862329741753387911, 1.93213126305064007059107443207, 2.95321747653964698216294685346, 3.03246709456858498475918707015, 3.12673878712068688196868780349, 3.22043251743417688051139364853, 3.66494612750793932267745417923, 3.97734129503180064768784414404, 4.05728632087198794572034964292, 4.37422959629813724442148635519, 5.00741716684383703557451421431, 5.12486970892762200839904860534, 5.51621562978604837944520528828, 5.62259359932216222501328682576, 5.91417708724242520864586791079, 5.99838225634515335205206757929, 6.02197424421362615696617875634, 6.82178126111040023748214800129, 6.84250419727852991433220720269, 6.91771776750852167465237574510, 7.28880710560473519556560070841, 7.38469421984987685418205477244, 7.84431998514043485500010396307
