| L(s) = 1 | + 3·5-s − 7-s + 11-s + 3·13-s + 17-s − 6·19-s − 7·23-s + 6·25-s − 18·29-s − 6·31-s − 3·35-s + 13·37-s − 41-s − 18·47-s − 4·49-s − 11·53-s + 3·55-s − 8·59-s + 9·61-s + 9·65-s − 4·67-s − 11·71-s − 6·73-s − 77-s − 5·79-s + 8·83-s + 3·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.301·11-s + 0.832·13-s + 0.242·17-s − 1.37·19-s − 1.45·23-s + 6/5·25-s − 3.34·29-s − 1.07·31-s − 0.507·35-s + 2.13·37-s − 0.156·41-s − 2.62·47-s − 4/7·49-s − 1.51·53-s + 0.404·55-s − 1.04·59-s + 1.15·61-s + 1.11·65-s − 0.488·67-s − 1.30·71-s − 0.702·73-s − 0.113·77-s − 0.562·79-s + 0.878·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 42 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 53 T^{2} + 194 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 646 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 91 T^{2} + 6 T^{3} + 91 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 18 T + 221 T^{2} + 1756 T^{3} + 221 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 167 T^{2} + 1162 T^{3} + 167 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 129 T^{2} + 816 T^{3} + 129 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 408 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 237 T^{2} + 1530 T^{3} + 237 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 532 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 854 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 201 T^{2} - 1200 T^{3} + 201 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 275 T^{2} + 1954 T^{3} + 275 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 467 T^{2} + 5094 T^{3} + 467 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.24797256344738364287926126492, −6.76529691164989551310313068313, −6.57669499194461959635170036677, −6.37729849763259450345597268643, −6.23811449562266395248880814810, −6.09024610070575552062979384903, −5.75239271112804467968828778615, −5.60417731988730913156134107572, −5.35547693496961393329091092477, −5.32149414825110179596937908779, −4.79271731615888810487528844481, −4.44427261807916518284124690327, −4.38491375563796596432008092389, −4.11296974696803624852310250561, −3.74060216563610477590844572368, −3.67886002200859448746017923744, −3.23980467055955888683695094546, −3.09777465896630303930439162194, −2.78150704409632401017580954555, −2.26598433142113130665580408162, −2.18684692440567046375887876874, −1.92914235376511877405223222358, −1.51245668206700871652279007050, −1.32410764524985695962530822765, −1.21651876374198224223291977250, 0, 0, 0,
1.21651876374198224223291977250, 1.32410764524985695962530822765, 1.51245668206700871652279007050, 1.92914235376511877405223222358, 2.18684692440567046375887876874, 2.26598433142113130665580408162, 2.78150704409632401017580954555, 3.09777465896630303930439162194, 3.23980467055955888683695094546, 3.67886002200859448746017923744, 3.74060216563610477590844572368, 4.11296974696803624852310250561, 4.38491375563796596432008092389, 4.44427261807916518284124690327, 4.79271731615888810487528844481, 5.32149414825110179596937908779, 5.35547693496961393329091092477, 5.60417731988730913156134107572, 5.75239271112804467968828778615, 6.09024610070575552062979384903, 6.23811449562266395248880814810, 6.37729849763259450345597268643, 6.57669499194461959635170036677, 6.76529691164989551310313068313, 7.24797256344738364287926126492
