| L(s) = 1 | − 3·2-s − 2·3-s + 6·4-s − 2·5-s + 6·6-s + 7-s − 10·8-s − 2·9-s + 6·10-s + 8·11-s − 12·12-s − 3·14-s + 4·15-s + 15·16-s − 2·17-s + 6·18-s − 3·19-s − 12·20-s − 2·21-s − 24·22-s + 2·23-s + 20·24-s − 8·25-s + 9·27-s + 6·28-s − 14·29-s − 12·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3·4-s − 0.894·5-s + 2.44·6-s + 0.377·7-s − 3.53·8-s − 2/3·9-s + 1.89·10-s + 2.41·11-s − 3.46·12-s − 0.801·14-s + 1.03·15-s + 15/4·16-s − 0.485·17-s + 1.41·18-s − 0.688·19-s − 2.68·20-s − 0.436·21-s − 5.11·22-s + 0.417·23-s + 4.08·24-s − 8/5·25-s + 1.73·27-s + 1.13·28-s − 2.59·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 13^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 13^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 7 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T + 12 T^{2} + 3 p T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - T + 17 T^{2} - 15 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 8 T + 50 T^{2} - 181 T^{3} + 50 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 2 T + 40 T^{2} - 87 T^{3} + 40 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 14 T + 122 T^{2} + 709 T^{3} + 122 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 45 T^{2} - 383 T^{3} + 45 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 59 T^{2} - 104 T^{3} + 59 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 15 T + 185 T^{2} + 1303 T^{3} + 185 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 120 T^{2} + 425 T^{3} + 120 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 11 T + 125 T^{2} - 1039 T^{3} + 125 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 5 T + 111 T^{2} - 603 T^{3} + 111 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 4 T + 48 T^{2} - 471 T^{3} + 48 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 2 T + 154 T^{2} - 161 T^{3} + 154 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 161 T^{2} + 812 T^{3} + 161 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 73 | $A_4\times C_2$ | \( 1 + 15 T + 86 T^{2} + 443 T^{3} + 86 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 2 T + 130 T^{2} + 545 T^{3} + 130 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 5 T - 7 T^{2} - 995 T^{3} - 7 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 27 T + 471 T^{2} - 5119 T^{3} + 471 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 20 T + 407 T^{2} - 4080 T^{3} + 407 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.55496993544870726255289889439, −7.31930416168031120948712560048, −6.85716497869784616587706242644, −6.79498640449527988252518285583, −6.51865175199154033935790104469, −6.45561807319838471800427829250, −6.27255760352584153492651933607, −5.73130386801674108895980679124, −5.68410609281123469787338892383, −5.59171317458142493492887726132, −5.15973124852144780475148022221, −4.79040806918424205654959949302, −4.60816654959569617834146911382, −4.08910573977145011158743393895, −3.92395533145189972565154304717, −3.88297935595348053280689834548, −3.32035582512412117948066313204, −3.12795236091458928767251397725, −3.06039045934108584464397455672, −2.10788266917865239000599187701, −2.10177373209538311474653444348, −2.00932972506099058329578799658, −1.49981887319571309617710356716, −1.04931985262467437840714382149, −0.943000181828108507209970197342, 0, 0, 0,
0.943000181828108507209970197342, 1.04931985262467437840714382149, 1.49981887319571309617710356716, 2.00932972506099058329578799658, 2.10177373209538311474653444348, 2.10788266917865239000599187701, 3.06039045934108584464397455672, 3.12795236091458928767251397725, 3.32035582512412117948066313204, 3.88297935595348053280689834548, 3.92395533145189972565154304717, 4.08910573977145011158743393895, 4.60816654959569617834146911382, 4.79040806918424205654959949302, 5.15973124852144780475148022221, 5.59171317458142493492887726132, 5.68410609281123469787338892383, 5.73130386801674108895980679124, 6.27255760352584153492651933607, 6.45561807319838471800427829250, 6.51865175199154033935790104469, 6.79498640449527988252518285583, 6.85716497869784616587706242644, 7.31930416168031120948712560048, 7.55496993544870726255289889439
