| L(s) = 1 | − 2·3-s − 4·5-s − 2·11-s − 4·13-s + 8·15-s + 6·17-s + 4·19-s − 4·23-s + 4·25-s + 4·27-s − 3·29-s − 14·31-s + 4·33-s + 2·37-s + 8·39-s + 10·41-s + 6·43-s − 2·47-s − 21·49-s − 12·51-s + 8·55-s − 8·57-s − 8·59-s − 2·61-s + 16·65-s − 20·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 0.603·11-s − 1.10·13-s + 2.06·15-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 4/5·25-s + 0.769·27-s − 0.557·29-s − 2.51·31-s + 0.696·33-s + 0.328·37-s + 1.28·39-s + 1.56·41-s + 0.914·43-s − 0.291·47-s − 3·49-s − 1.68·51-s + 1.07·55-s − 1.05·57-s − 1.04·59-s − 0.256·61-s + 1.98·65-s − 2.44·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 4 T^{2} + 4 T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 6 p T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 4 T^{2} - 36 T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 102 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 29 T^{2} - 120 T^{3} + 29 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 152 T^{2} + 936 T^{3} + 152 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 79 T^{2} - 116 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 59 T^{2} - 308 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 92 T^{2} - 548 T^{3} + 92 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 24 T^{2} - 264 T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 116 T^{2} - 58 T^{3} + 116 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 173 T^{2} + 928 T^{3} + 173 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 83 T^{2} - 84 T^{3} + 83 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 233 T^{2} + 2040 T^{3} + 233 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 65 T^{2} - 8 T^{3} + 65 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 123 T^{2} - 452 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 30 T + 488 T^{2} + 5128 T^{3} + 488 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 32 T + 565 T^{2} + 6288 T^{3} + 565 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 11 T^{2} + 1036 T^{3} + 11 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 323 T^{2} + 2652 T^{3} + 323 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.661277786226985694462494277155, −8.140692090360673915117366412993, −7.88076835612838016225791722496, −7.66835227500665221258435600384, −7.66274759331615056564596168996, −7.35415163373785081226255233041, −7.22119742503973344024085547403, −6.78605823856849712101488452154, −6.44109397423777392220804178542, −6.04997363806133720166089019663, −5.77988100972354318878948893235, −5.63451580798356693331713996935, −5.34987612363974656108832826102, −5.18959555993701630854912359154, −4.74030504469101430973098537348, −4.43970306943864635980738033451, −4.05279742046282290633945593517, −4.02249764325837518532076433960, −3.52536742096617600146168597393, −3.13454810722372892081693506686, −2.90025256629270214147122119200, −2.68064428940024873700939033743, −1.99183556176515530504696461667, −1.39338316208286742877005505906, −1.24264055230817151923655035775, 0, 0, 0,
1.24264055230817151923655035775, 1.39338316208286742877005505906, 1.99183556176515530504696461667, 2.68064428940024873700939033743, 2.90025256629270214147122119200, 3.13454810722372892081693506686, 3.52536742096617600146168597393, 4.02249764325837518532076433960, 4.05279742046282290633945593517, 4.43970306943864635980738033451, 4.74030504469101430973098537348, 5.18959555993701630854912359154, 5.34987612363974656108832826102, 5.63451580798356693331713996935, 5.77988100972354318878948893235, 6.04997363806133720166089019663, 6.44109397423777392220804178542, 6.78605823856849712101488452154, 7.22119742503973344024085547403, 7.35415163373785081226255233041, 7.66274759331615056564596168996, 7.66835227500665221258435600384, 7.88076835612838016225791722496, 8.140692090360673915117366412993, 8.661277786226985694462494277155
