| L(s) = 1 | + 3·2-s − 3·3-s + 7·4-s + 6·5-s − 9·6-s + 2·7-s + 14·8-s + 6·9-s + 18·10-s + 5·11-s − 21·12-s + 6·14-s − 18·15-s + 21·16-s − 17-s + 18·18-s − 7·19-s + 42·20-s − 6·21-s + 15·22-s − 42·24-s + 16·25-s − 10·27-s + 14·28-s − 2·29-s − 54·30-s − 16·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 7/2·4-s + 2.68·5-s − 3.67·6-s + 0.755·7-s + 4.94·8-s + 2·9-s + 5.69·10-s + 1.50·11-s − 6.06·12-s + 1.60·14-s − 4.64·15-s + 21/4·16-s − 0.242·17-s + 4.24·18-s − 1.60·19-s + 9.39·20-s − 1.30·21-s + 3.19·22-s − 8.57·24-s + 16/5·25-s − 1.92·27-s + 2.64·28-s − 0.371·29-s − 9.85·30-s − 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.26539294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.26539294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 - 3 T + p T^{2} + T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 47 T^{3} + 4 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 2 T + 20 T^{2} - 27 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 69 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 7 T + 71 T^{2} + 273 T^{3} + 71 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} + 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 16 T + 134 T^{2} + 795 T^{3} + 134 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 22 T + 270 T^{2} - 2005 T^{3} + 270 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 11 T + 147 T^{2} - 873 T^{3} + 147 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 15 T + 176 T^{2} + 1331 T^{3} + 176 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 155 T^{2} - 665 T^{3} + 155 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 161 T^{2} + 604 T^{3} + 161 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 11 T + 155 T^{2} - 1515 T^{3} + 155 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 122 T^{2} + 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 84 T^{2} + 47 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 12 T + 290 T^{2} - 2035 T^{3} + 290 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - T + 167 T^{2} - 65 T^{3} + 167 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 10 T^{2} - 667 T^{3} + 10 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.718590993472648888920182265333, −9.541523968179841066512171474452, −9.365669293112238223844295430788, −9.207476267830635728608549577461, −8.506509273980416654680785872432, −7.931350372598345093761783796969, −7.69722221732747243271926069083, −7.34056233531221372370716228875, −6.82303400118216542485672566361, −6.49836467154880290748927830507, −6.45385532909737390433362921761, −6.13159882277292972167793745591, −6.08007829497662988317020565119, −5.58778236919420814783250669715, −5.34355285688175209912160995369, −5.06275700736903279327421974501, −4.47555183432868305616718971206, −4.31631134428115154031449675131, −4.29783117408011316016719717866, −3.41217922415537127594478474991, −3.00726401005179861311751062407, −2.05843232776131367445206750053, −1.96556914197670966460621889314, −1.87187510316131161114710467104, −1.25195005858148234988264507219,
1.25195005858148234988264507219, 1.87187510316131161114710467104, 1.96556914197670966460621889314, 2.05843232776131367445206750053, 3.00726401005179861311751062407, 3.41217922415537127594478474991, 4.29783117408011316016719717866, 4.31631134428115154031449675131, 4.47555183432868305616718971206, 5.06275700736903279327421974501, 5.34355285688175209912160995369, 5.58778236919420814783250669715, 6.08007829497662988317020565119, 6.13159882277292972167793745591, 6.45385532909737390433362921761, 6.49836467154880290748927830507, 6.82303400118216542485672566361, 7.34056233531221372370716228875, 7.69722221732747243271926069083, 7.931350372598345093761783796969, 8.506509273980416654680785872432, 9.207476267830635728608549577461, 9.365669293112238223844295430788, 9.541523968179841066512171474452, 9.718590993472648888920182265333
