| L(s) = 1 | − 2-s + 2·3-s − 4-s + 2·5-s − 2·6-s + 8-s + 9-s − 2·10-s − 2·11-s − 2·12-s + 4·15-s − 16-s − 4·17-s − 18-s − 4·19-s − 2·20-s + 2·22-s + 10·23-s + 2·24-s − 8·25-s + 24·29-s − 4·30-s − 4·31-s + 32-s − 4·33-s + 4·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.577·12-s + 1.03·15-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.426·22-s + 2.08·23-s + 0.408·24-s − 8/5·25-s + 4.45·29-s − 0.730·30-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + 0.685·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.529370336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529370336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 12 T^{2} - 18 T^{3} + 12 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 36 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 58 T^{2} + 148 T^{3} + 58 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 70 T^{2} - 324 T^{3} + 70 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 74 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 95 T^{2} - 172 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 62 T^{2} + 208 T^{3} + 62 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 21 T^{2} - 216 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 868 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 120 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T - 22 T^{2} - 1276 T^{3} - 22 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 172 T^{2} + 66 T^{3} + 172 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 320 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.99513608344172690540322173047, −6.72813126981089345010888097907, −6.38491433425356465756221500644, −6.28632242903058509094775254495, −6.15553634445176828404268341318, −5.65502527010442978976445791935, −5.53942925144540527765526344285, −5.19612282135210790628057084835, −4.95430035750646220004965193425, −4.87253910406839158401923624218, −4.32925584513682321414595936974, −4.28990263982267676262659249156, −4.14028191921784606671357751026, −3.83346637585511154096514918017, −3.24412042499830330446531350310, −3.16595504086723249201992777847, −2.80935165752218388585585009821, −2.56495374518596115872729003148, −2.38075751983660500864425938109, −2.34412177363160750335198883222, −1.74110741684718961817209880768, −1.29412056216131083000370105970, −1.26038817222749649421211644480, −0.60997768891700025543239053323, −0.31546637797499390687043156064,
0.31546637797499390687043156064, 0.60997768891700025543239053323, 1.26038817222749649421211644480, 1.29412056216131083000370105970, 1.74110741684718961817209880768, 2.34412177363160750335198883222, 2.38075751983660500864425938109, 2.56495374518596115872729003148, 2.80935165752218388585585009821, 3.16595504086723249201992777847, 3.24412042499830330446531350310, 3.83346637585511154096514918017, 4.14028191921784606671357751026, 4.28990263982267676262659249156, 4.32925584513682321414595936974, 4.87253910406839158401923624218, 4.95430035750646220004965193425, 5.19612282135210790628057084835, 5.53942925144540527765526344285, 5.65502527010442978976445791935, 6.15553634445176828404268341318, 6.28632242903058509094775254495, 6.38491433425356465756221500644, 6.72813126981089345010888097907, 6.99513608344172690540322173047
