| L(s) = 1 | + 2-s − 5·7-s + 2·11-s − 4·13-s − 5·14-s − 16-s − 2·17-s + 4·19-s + 2·22-s + 3·23-s − 4·26-s + 7·29-s + 8·31-s + 32-s − 2·34-s − 6·37-s + 4·38-s + 13·41-s − 10·43-s + 3·46-s + 13·47-s + 49-s − 2·53-s + 7·58-s + 2·59-s + 61-s + 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.88·7-s + 0.603·11-s − 1.10·13-s − 1.33·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.426·22-s + 0.625·23-s − 0.784·26-s + 1.29·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.986·37-s + 0.648·38-s + 2.03·41-s − 1.52·43-s + 0.442·46-s + 1.89·47-s + 1/7·49-s − 0.274·53-s + 0.919·58-s + 0.260·59-s + 0.128·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.330276616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.330276616\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T + 24 T^{2} + 67 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 32 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 100 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 56 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 36 T^{2} - 21 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 2 p T^{2} - 355 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 28 T^{3} + 33 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 440 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 142 T^{2} - 1069 T^{3} + 142 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 856 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 130 T^{2} - 853 T^{3} + 130 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 139 T^{2} + 188 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 157 T^{2} - 212 T^{3} + 157 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 146 T^{2} - 193 T^{3} + 146 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 11 T + 162 T^{2} + 967 T^{3} + 162 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 121 T^{2} - 712 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 155 T^{2} - 1040 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 153 T^{2} - 340 T^{3} + 153 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 276 T^{2} + 2409 T^{3} + 276 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 255 T^{2} - 2188 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−8.211527110233392544682125593207, −7.77669608549817457370033162188, −7.53531805873950432305533057147, −7.13902986112849766477001504533, −7.04524175467442871477922073996, −6.69209058393316215458952147572, −6.49500507722662971731118198212, −6.36249603297419901403594139859, −5.97702706078767304234742622352, −5.77725380322262836118724774880, −5.33851932071977532698635648259, −5.06066603766496028718561859416, −4.83837113944155071729613140099, −4.40034036081797395495753319077, −4.37853712658312925179255742109, −4.00708330516824770996570925111, −3.47654433051217968674212814143, −3.26977644398668818177130197272, −3.06894169250022762382169061074, −2.61217837855534947902509196977, −2.55463989733220020119026804596, −1.92603977633517897054682920440, −1.44682668152837115788130489290, −0.67598102232932628298150578290, −0.55987360486332355815705329965,
0.55987360486332355815705329965, 0.67598102232932628298150578290, 1.44682668152837115788130489290, 1.92603977633517897054682920440, 2.55463989733220020119026804596, 2.61217837855534947902509196977, 3.06894169250022762382169061074, 3.26977644398668818177130197272, 3.47654433051217968674212814143, 4.00708330516824770996570925111, 4.37853712658312925179255742109, 4.40034036081797395495753319077, 4.83837113944155071729613140099, 5.06066603766496028718561859416, 5.33851932071977532698635648259, 5.77725380322262836118724774880, 5.97702706078767304234742622352, 6.36249603297419901403594139859, 6.49500507722662971731118198212, 6.69209058393316215458952147572, 7.04524175467442871477922073996, 7.13902986112849766477001504533, 7.53531805873950432305533057147, 7.77669608549817457370033162188, 8.211527110233392544682125593207
