| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s − 8·11-s + 4·13-s + 15·16-s − 2·17-s + 3·19-s + 24·22-s + 8·23-s − 12·26-s − 10·29-s − 21·32-s + 6·34-s + 4·37-s − 9·38-s − 4·41-s − 6·43-s − 48·44-s − 24·46-s − 8·47-s − 5·49-s + 24·52-s + 18·53-s + 30·58-s − 10·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 3.53·8-s − 2.41·11-s + 1.10·13-s + 15/4·16-s − 0.485·17-s + 0.688·19-s + 5.11·22-s + 1.66·23-s − 2.35·26-s − 1.85·29-s − 3.71·32-s + 1.02·34-s + 0.657·37-s − 1.45·38-s − 0.624·41-s − 0.914·43-s − 7.23·44-s − 3.53·46-s − 1.16·47-s − 5/7·49-s + 3.32·52-s + 2.47·53-s + 3.93·58-s − 1.30·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288723764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288723764\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 160 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 72 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} + 60 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 572 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 43 T^{2} - 72 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 244 T^{3} + 5 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 149 T^{2} + 736 T^{3} + 149 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 173 T^{2} + 1172 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 20 T + 249 T^{2} - 2360 T^{3} + 249 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 20 T + 261 T^{2} + 2520 T^{3} + 261 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 155 T^{2} - 912 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 221 T^{2} - 16 T^{3} + 221 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 504 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 24 T + 443 T^{2} + 4672 T^{3} + 443 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 439 T^{2} - 4564 T^{3} + 439 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96868488315335213408946055544, −6.67199170811667325021736370539, −6.62487777698329249771059391991, −6.47324647198573705915022576547, −5.79047488542164358393696223561, −5.75643926846827802732727431159, −5.63019648636041995016179304690, −5.36046734424467050629469557726, −5.04839638533060583006422556797, −5.02521117219958121031058577034, −4.48585639664261130236534176268, −4.25368753709149120197544486505, −3.90853117734720178101251036931, −3.46286787526812040789781795366, −3.35943424569274292192689227386, −3.11902141056990383847453744543, −2.76576253754938955573947207686, −2.56409794199333565931121708535, −2.35683643481579403163961841076, −1.79581902305663366305092530818, −1.65570175032395001083512406353, −1.56373350943480798128875077943, −0.77727258028420918011490586216, −0.53372213282083449528426349315, −0.44685535770629483631864430847,
0.44685535770629483631864430847, 0.53372213282083449528426349315, 0.77727258028420918011490586216, 1.56373350943480798128875077943, 1.65570175032395001083512406353, 1.79581902305663366305092530818, 2.35683643481579403163961841076, 2.56409794199333565931121708535, 2.76576253754938955573947207686, 3.11902141056990383847453744543, 3.35943424569274292192689227386, 3.46286787526812040789781795366, 3.90853117734720178101251036931, 4.25368753709149120197544486505, 4.48585639664261130236534176268, 5.02521117219958121031058577034, 5.04839638533060583006422556797, 5.36046734424467050629469557726, 5.63019648636041995016179304690, 5.75643926846827802732727431159, 5.79047488542164358393696223561, 6.47324647198573705915022576547, 6.62487777698329249771059391991, 6.67199170811667325021736370539, 6.96868488315335213408946055544
