| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 4·5-s + 9·6-s − 3·7-s − 10·8-s + 6·9-s − 12·10-s − 3·11-s − 18·12-s + 9·14-s − 12·15-s + 15·16-s − 3·17-s − 18·18-s + 7·19-s + 24·20-s + 9·21-s + 9·22-s − 5·23-s + 30·24-s − 2·25-s − 10·27-s − 18·28-s − 2·29-s + 36·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 1.78·5-s + 3.67·6-s − 1.13·7-s − 3.53·8-s + 2·9-s − 3.79·10-s − 0.904·11-s − 5.19·12-s + 2.40·14-s − 3.09·15-s + 15/4·16-s − 0.727·17-s − 4.24·18-s + 1.60·19-s + 5.36·20-s + 1.96·21-s + 1.91·22-s − 1.04·23-s + 6.12·24-s − 2/5·25-s − 1.92·27-s − 3.40·28-s − 0.371·29-s + 6.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4800883465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4800883465\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 4 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 39 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 103 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 + 5 T + 33 T^{2} + 273 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 44 T^{2} - 11 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 10 T + 96 T^{2} - 607 T^{3} + 96 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 104 T^{2} - 7 T^{3} + 104 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 4 T + 126 T^{2} + 327 T^{3} + 126 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 7 T + 101 T^{2} - 399 T^{3} + 101 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 116 T^{2} - 199 T^{3} + 116 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 2 T + 88 T^{2} + 99 T^{3} + 88 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 2 T + 148 T^{2} + 249 T^{3} + 148 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{3} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 122 T^{2} - 343 T^{3} + 122 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 276 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 11 T + 243 T^{2} - 1577 T^{3} + 243 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 22 T + 396 T^{2} + 3853 T^{3} + 396 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 31 T + 567 T^{2} - 75 p T^{3} + 567 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T^{2} + 889 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 28 T + 536 T^{2} - 63 p T^{3} + 536 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.18390564776536487157838539598, −6.54521858468064413359049184831, −6.52514325212672993968322131829, −6.41384287983191603786852875839, −6.14160838111282070179804425311, −5.89568052715302968850076888422, −5.87809523872014249488017770734, −5.40133812151791679144509938890, −5.30876401176416079512635484369, −5.28899718937192495046509167177, −4.54070778538345224235795410783, −4.40616886268087405846747658511, −4.38477249566000423743941385730, −3.50252655809650479940994392574, −3.48754243512050748340736826279, −3.33198026447704980326722083586, −2.52730519173745184241719212224, −2.51114319266458694681130378416, −2.50970984463336526295708618593, −1.75808673959642332915898939202, −1.66334415309830203728194267935, −1.55354456807299721256433767627, −0.73324093869823504627329628056, −0.69513225540591723354373854136, −0.27578105648960445482675389353,
0.27578105648960445482675389353, 0.69513225540591723354373854136, 0.73324093869823504627329628056, 1.55354456807299721256433767627, 1.66334415309830203728194267935, 1.75808673959642332915898939202, 2.50970984463336526295708618593, 2.51114319266458694681130378416, 2.52730519173745184241719212224, 3.33198026447704980326722083586, 3.48754243512050748340736826279, 3.50252655809650479940994392574, 4.38477249566000423743941385730, 4.40616886268087405846747658511, 4.54070778538345224235795410783, 5.28899718937192495046509167177, 5.30876401176416079512635484369, 5.40133812151791679144509938890, 5.87809523872014249488017770734, 5.89568052715302968850076888422, 6.14160838111282070179804425311, 6.41384287983191603786852875839, 6.52514325212672993968322131829, 6.54521858468064413359049184831, 7.18390564776536487157838539598
