| L(s) = 1 | + 2·2-s + 4-s − 4·7-s − 8-s − 11-s − 3·13-s − 8·14-s − 5·16-s + 14·17-s + 3·19-s − 2·22-s + 8·23-s − 6·26-s − 4·28-s + 5·29-s − 31-s − 8·32-s + 28·34-s − 5·37-s + 6·38-s − 41-s + 5·43-s − 44-s + 16·46-s + 9·47-s − 6·49-s − 3·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 0.301·11-s − 0.832·13-s − 2.13·14-s − 5/4·16-s + 3.39·17-s + 0.688·19-s − 0.426·22-s + 1.66·23-s − 1.17·26-s − 0.755·28-s + 0.928·29-s − 0.179·31-s − 1.41·32-s + 4.80·34-s − 0.821·37-s + 0.973·38-s − 0.156·41-s + 0.762·43-s − 0.150·44-s + 2.35·46-s + 1.31·47-s − 6/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.374817631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.374817631\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 22 T^{2} + 55 T^{3} + 22 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 29 T^{2} + 23 T^{3} + 29 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 25 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 14 T + 112 T^{2} - 555 T^{3} + 112 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 8 T + 60 T^{2} - 243 T^{3} + 60 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 5 T + 91 T^{2} - 285 T^{3} + 91 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 115 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + p T^{2} - 25 T^{3} + p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + p T^{2} - 73 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 16 T^{2} - 113 T^{3} + 16 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 77 T^{2} - 535 T^{3} + 77 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 31 T + 462 T^{2} - 4191 T^{3} + 462 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 508 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 1573 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 212 T^{2} + 947 T^{3} + 212 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T + 189 T^{2} + 199 T^{3} + 189 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 254 T^{2} + 31 p T^{3} + 254 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 3 T - 21 T^{2} - 629 T^{3} - 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 20 T + 318 T^{2} - 3435 T^{3} + 318 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 2353 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.30848830419549750562486742258, −7.20299626456469796822449563957, −7.01937045931268177550805129642, −6.57783274606217489612406022962, −6.48708038096055355562751741547, −5.98327309663853387818180422712, −5.85014867839437979829095057142, −5.48402081525655015474837100765, −5.45973485174787373467421880379, −5.30362904113739979880436818796, −4.81257645616887778638140445208, −4.68021651537084873304692788291, −4.59537704372274093724485294848, −3.82429276756085675053713750208, −3.71316498053751411295420668696, −3.66535353405405435315061388968, −3.40993454429543777336460815027, −2.96331046239774651272285782731, −2.75530088610732733393894131303, −2.51629820984934939324235352370, −2.31121733241419916104656973415, −1.57022497673754590660678124160, −1.15065948936924680686550817808, −0.73767620475483239462312260767, −0.51205925696299163643904050997,
0.51205925696299163643904050997, 0.73767620475483239462312260767, 1.15065948936924680686550817808, 1.57022497673754590660678124160, 2.31121733241419916104656973415, 2.51629820984934939324235352370, 2.75530088610732733393894131303, 2.96331046239774651272285782731, 3.40993454429543777336460815027, 3.66535353405405435315061388968, 3.71316498053751411295420668696, 3.82429276756085675053713750208, 4.59537704372274093724485294848, 4.68021651537084873304692788291, 4.81257645616887778638140445208, 5.30362904113739979880436818796, 5.45973485174787373467421880379, 5.48402081525655015474837100765, 5.85014867839437979829095057142, 5.98327309663853387818180422712, 6.48708038096055355562751741547, 6.57783274606217489612406022962, 7.01937045931268177550805129642, 7.20299626456469796822449563957, 7.30848830419549750562486742258
