| L(s) = 1 | + 3-s + 5-s − 2·7-s − 9-s + 4·11-s − 13-s + 15-s + 11·17-s − 2·21-s + 23-s − 10·25-s + 4·27-s − 3·29-s − 6·31-s + 4·33-s − 2·35-s − 12·37-s − 39-s − 19·41-s + 5·43-s − 45-s + 17·47-s − 3·49-s + 11·51-s − 5·53-s + 4·55-s + 13·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s − 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 2.66·17-s − 0.436·21-s + 0.208·23-s − 2·25-s + 0.769·27-s − 0.557·29-s − 1.07·31-s + 0.696·33-s − 0.338·35-s − 1.97·37-s − 0.160·39-s − 2.96·41-s + 0.762·43-s − 0.149·45-s + 2.47·47-s − 3/7·49-s + 1.54·51-s − 0.686·53-s + 0.539·55-s + 1.69·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7427352126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7427352126\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} - 7 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 34 T^{2} - 84 T^{3} + 34 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 7 T^{2} - 50 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 75 T^{2} - 342 T^{3} + 75 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 65 T^{2} - 44 T^{3} + 65 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 176 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 39 T^{2} + 156 T^{3} + 39 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 93 T^{2} + 596 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 19 T + 214 T^{2} + 39 p T^{3} + 214 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 5 T + 45 T^{2} - 2 p T^{3} + 45 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 17 T + 3 p T^{2} - 876 T^{3} + 3 p^{2} T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 5 T + 151 T^{2} + 486 T^{3} + 151 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 13 T + 226 T^{2} - 1587 T^{3} + 226 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} + 608 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 170 T^{2} + 1183 T^{3} + 170 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 209 T^{2} + 422 T^{3} + 209 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 11 T + 242 T^{2} + 1587 T^{3} + 242 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 19 T + 325 T^{2} + 2986 T^{3} + 325 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 206 T^{2} - 1360 T^{3} + 206 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 83 T^{2} + 10 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - T + 274 T^{2} - 193 T^{3} + 274 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79578444146687154633985290496, −7.52889964029252082406694232326, −7.24251204551751573295572654712, −7.13010052319320668667941580425, −6.68189364739947068735242823568, −6.56627651566096033756357975462, −6.28863265323901321591171644067, −5.86072542263221445190859085501, −5.65081038553404361773733689542, −5.53936822567658684887672069347, −5.17394061017284096222034441431, −5.15481359527628758950317303555, −4.55047827049431742796931243021, −4.02530999882175080464096884737, −3.98147961842964076630950390066, −3.74198118677421519650237278523, −3.30930387187050434400058877868, −3.13011180503176896545404690110, −3.00279507739663639574548361722, −2.45766953413315737078433906698, −2.01346908494270965761317086639, −1.76512126888006407099374772359, −1.25755954460316657228148922143, −1.15897013414195697438185597817, −0.15983361109685320971249121864,
0.15983361109685320971249121864, 1.15897013414195697438185597817, 1.25755954460316657228148922143, 1.76512126888006407099374772359, 2.01346908494270965761317086639, 2.45766953413315737078433906698, 3.00279507739663639574548361722, 3.13011180503176896545404690110, 3.30930387187050434400058877868, 3.74198118677421519650237278523, 3.98147961842964076630950390066, 4.02530999882175080464096884737, 4.55047827049431742796931243021, 5.15481359527628758950317303555, 5.17394061017284096222034441431, 5.53936822567658684887672069347, 5.65081038553404361773733689542, 5.86072542263221445190859085501, 6.28863265323901321591171644067, 6.56627651566096033756357975462, 6.68189364739947068735242823568, 7.13010052319320668667941580425, 7.24251204551751573295572654712, 7.52889964029252082406694232326, 7.79578444146687154633985290496
