| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 9·6-s + 10·8-s + 6·9-s − 4·11-s + 18·12-s + 6·13-s + 15·16-s + 10·17-s + 18·18-s − 3·19-s − 12·22-s + 6·23-s + 30·24-s + 18·26-s + 10·27-s + 10·29-s + 14·31-s + 21·32-s − 12·33-s + 30·34-s + 36·36-s + 2·37-s − 9·38-s + 18·39-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s + 3.67·6-s + 3.53·8-s + 2·9-s − 1.20·11-s + 5.19·12-s + 1.66·13-s + 15/4·16-s + 2.42·17-s + 4.24·18-s − 0.688·19-s − 2.55·22-s + 1.25·23-s + 6.12·24-s + 3.53·26-s + 1.92·27-s + 1.85·29-s + 2.51·31-s + 3.71·32-s − 2.08·33-s + 5.14·34-s + 6·36-s + 0.328·37-s − 1.45·38-s + 2.88·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(54.85584256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(54.85584256\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 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 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 71 T^{2} - 332 T^{3} + 71 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 65 T^{2} - 268 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 121 T^{2} - 716 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 380 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 107 T^{2} - 16 T^{3} + 107 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 852 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 89 T^{2} + 4 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 171 T^{2} - 1332 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 229 T^{2} - 1692 T^{3} + 229 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 133 T^{2} + 504 T^{3} + 133 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 22 T + 361 T^{2} - 3676 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} + 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 219 T^{2} - 792 T^{3} + 219 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 111 T^{2} + 948 T^{3} + 111 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.931478396914411953323697427228, −7.48613069896092329285770623366, −7.11150461525380051591678627816, −7.10601437125796402543833101237, −6.56115326544906252245241438575, −6.45560551033771189035124495192, −6.43045289009265658999750085701, −5.78133640412975049654380276877, −5.66936420649730443435582694185, −5.43787587012584236281882207117, −4.92973724455322693264210999141, −4.87725343105415445933195318676, −4.71688758940692180821682921486, −4.06505954267429588213190028360, −4.02769365733197114508109859252, −3.80000799176177355598201842257, −3.25134054008919156790089455244, −3.13703559769369937714758083238, −3.04491240443393951626008519290, −2.67580606341784920537303206430, −2.24778251805897735242666201997, −2.21896005726333935424128097692, −1.25504898012986686130007876283, −1.23499742185961202874085860454, −0.959314908362890965376886831490,
0.959314908362890965376886831490, 1.23499742185961202874085860454, 1.25504898012986686130007876283, 2.21896005726333935424128097692, 2.24778251805897735242666201997, 2.67580606341784920537303206430, 3.04491240443393951626008519290, 3.13703559769369937714758083238, 3.25134054008919156790089455244, 3.80000799176177355598201842257, 4.02769365733197114508109859252, 4.06505954267429588213190028360, 4.71688758940692180821682921486, 4.87725343105415445933195318676, 4.92973724455322693264210999141, 5.43787587012584236281882207117, 5.66936420649730443435582694185, 5.78133640412975049654380276877, 6.43045289009265658999750085701, 6.45560551033771189035124495192, 6.56115326544906252245241438575, 7.10601437125796402543833101237, 7.11150461525380051591678627816, 7.48613069896092329285770623366, 7.931478396914411953323697427228
