| L(s) = 1 | + 3-s + 2·7-s + 9-s − 7·11-s + 13-s − 10·17-s + 13·19-s + 2·21-s + 3·23-s − 27-s + 13·29-s + 8·31-s − 7·33-s − 5·37-s + 39-s + 8·41-s + 24·43-s + 2·47-s − 9·49-s − 10·51-s − 53-s + 13·57-s + 17·59-s + 13·61-s + 2·63-s + 5·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 2.11·11-s + 0.277·13-s − 2.42·17-s + 2.98·19-s + 0.436·21-s + 0.625·23-s − 0.192·27-s + 2.41·29-s + 1.43·31-s − 1.21·33-s − 0.821·37-s + 0.160·39-s + 1.24·41-s + 3.65·43-s + 0.291·47-s − 9/7·49-s − 1.40·51-s − 0.137·53-s + 1.72·57-s + 2.21·59-s + 1.66·61-s + 0.251·63-s + 0.610·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.788233683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.788233683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 27 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 40 T^{2} + 140 T^{3} + 40 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T + 16 T^{2} + 24 T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 61 T^{2} + 257 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 13 T + 90 T^{2} - 432 T^{3} + 90 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 113 T^{2} - 678 T^{3} + 113 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 105 T^{2} - 495 T^{3} + 105 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 19 T^{2} - 126 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 49 T^{2} + 212 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + T + 59 T^{2} + 406 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 17 T + 243 T^{2} - 2042 T^{3} + 243 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 132 T^{2} - 866 T^{3} + 132 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 109 T^{2} - 174 T^{3} + 109 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 22 T + 309 T^{2} - 2899 T^{3} + 309 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 43 T^{2} - 368 T^{3} + 43 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 120 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 289 T^{2} + 2454 T^{3} + 289 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 210 T^{2} - 632 T^{3} + 210 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.04461633411761207641521195176, −6.55958027722379497890792744090, −6.38577104640094956224100805694, −6.21744034304551447475109766317, −5.67513241107019343116764867238, −5.63154062862686269885341364263, −5.56245859420198612273231939676, −4.97117943518970933271121890751, −4.96935088247515494532900204263, −4.78017624992553538080769434444, −4.54438445145146960783571510516, −4.30823226548199334493005187115, −3.94633427184926664541625160908, −3.77571440525034708741688765385, −3.29631879390504821323441081147, −3.11361410404870560773545844973, −2.77301195451589434801571509523, −2.67674998228281818879399365673, −2.39726588774013309314386967935, −2.12710826457702905560000650039, −1.96628194540340173027771409700, −1.33831345209893637612313925070, −0.838053412475969857614892479209, −0.791786886986594595732976553433, −0.56084878313414926616442661496,
0.56084878313414926616442661496, 0.791786886986594595732976553433, 0.838053412475969857614892479209, 1.33831345209893637612313925070, 1.96628194540340173027771409700, 2.12710826457702905560000650039, 2.39726588774013309314386967935, 2.67674998228281818879399365673, 2.77301195451589434801571509523, 3.11361410404870560773545844973, 3.29631879390504821323441081147, 3.77571440525034708741688765385, 3.94633427184926664541625160908, 4.30823226548199334493005187115, 4.54438445145146960783571510516, 4.78017624992553538080769434444, 4.96935088247515494532900204263, 4.97117943518970933271121890751, 5.56245859420198612273231939676, 5.63154062862686269885341364263, 5.67513241107019343116764867238, 6.21744034304551447475109766317, 6.38577104640094956224100805694, 6.55958027722379497890792744090, 7.04461633411761207641521195176
