| L(s) = 1 | − 4·3-s − 2·7-s + 5·9-s + 6·11-s − 3·13-s + 6·17-s + 8·21-s − 14·23-s + 2·27-s + 6·29-s + 10·31-s − 24·33-s + 12·39-s − 4·41-s − 6·43-s − 10·47-s − 13·49-s − 24·51-s + 8·53-s + 8·59-s + 6·61-s − 10·63-s − 10·67-s + 56·69-s + 12·71-s − 24·73-s − 12·77-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.755·7-s + 5/3·9-s + 1.80·11-s − 0.832·13-s + 1.45·17-s + 1.74·21-s − 2.91·23-s + 0.384·27-s + 1.11·29-s + 1.79·31-s − 4.17·33-s + 1.92·39-s − 0.624·41-s − 0.914·43-s − 1.45·47-s − 1.85·49-s − 3.36·51-s + 1.09·53-s + 1.04·59-s + 0.768·61-s − 1.25·63-s − 1.22·67-s + 6.74·69-s + 1.42·71-s − 2.80·73-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T + 11 T^{2} + 22 T^{3} + 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 24 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 130 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 196 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 2 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 131 T^{2} + 730 T^{3} + 131 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 240 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 113 T^{2} - 594 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 83 T^{2} + 52 T^{3} + 83 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 91 T^{2} + 360 T^{3} + 91 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 238 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 169 T^{2} + 960 T^{3} + 169 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 119 T^{2} - 544 T^{3} + 119 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 137 T^{2} - 682 T^{3} + 137 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 167 T^{2} - 736 T^{3} + 167 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 141 T^{2} + 736 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 125 T^{2} - 950 T^{3} + 125 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 24 T + 383 T^{2} + 3740 T^{3} + 383 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 261 T^{2} - 2512 T^{3} + 261 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 22 T + 401 T^{2} + 3968 T^{3} + 401 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 215 T^{2} - 1580 T^{3} + 215 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 207 T^{2} + 2516 T^{3} + 207 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.64523169440409604579563296798, −7.14040692597489347706798136075, −6.90577344860335913316524743225, −6.82126401956356821762646584956, −6.42100800592798202217411094099, −6.42098711109495526500029797985, −6.26796942265983479772136612110, −5.92606036840446775528011652234, −5.74419057400454728161333764669, −5.51361624972159369290565204195, −5.17359077625479052982552580477, −5.08876333444008893535723820287, −4.82528727490835724685247217802, −4.31999849267180540402255444578, −4.15813411828844895999422831895, −4.07639020145742779706931558972, −3.61983048914564714792699826258, −3.23850140314274897215862785596, −3.20273345362771851962423778125, −2.57180967806542904623688944983, −2.48965514498331128132868116726, −1.97899184605421126409258205670, −1.40135796890742127800191959454, −1.18737548935082675442780730972, −1.06977528171549082018292440712, 0, 0, 0,
1.06977528171549082018292440712, 1.18737548935082675442780730972, 1.40135796890742127800191959454, 1.97899184605421126409258205670, 2.48965514498331128132868116726, 2.57180967806542904623688944983, 3.20273345362771851962423778125, 3.23850140314274897215862785596, 3.61983048914564714792699826258, 4.07639020145742779706931558972, 4.15813411828844895999422831895, 4.31999849267180540402255444578, 4.82528727490835724685247217802, 5.08876333444008893535723820287, 5.17359077625479052982552580477, 5.51361624972159369290565204195, 5.74419057400454728161333764669, 5.92606036840446775528011652234, 6.26796942265983479772136612110, 6.42098711109495526500029797985, 6.42100800592798202217411094099, 6.82126401956356821762646584956, 6.90577344860335913316524743225, 7.14040692597489347706798136075, 7.64523169440409604579563296798
