| L(s) = 1 | − 2·3-s + 4·7-s − 9-s + 2·11-s + 8·13-s + 4·17-s + 2·19-s − 8·21-s + 8·23-s + 4·27-s − 6·29-s − 8·31-s − 4·33-s + 8·37-s − 16·39-s + 2·41-s − 14·43-s + 4·47-s − 49-s − 8·51-s + 16·53-s − 4·57-s − 2·59-s − 10·61-s − 4·63-s − 10·67-s − 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s − 1/3·9-s + 0.603·11-s + 2.21·13-s + 0.970·17-s + 0.458·19-s − 1.74·21-s + 1.66·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 1.31·37-s − 2.56·39-s + 0.312·41-s − 2.13·43-s + 0.583·47-s − 1/7·49-s − 1.12·51-s + 2.19·53-s − 0.529·57-s − 0.260·59-s − 1.28·61-s − 0.503·63-s − 1.22·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.032020420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.032020420\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 36 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 104 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} + 28 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 81 T^{2} - 372 T^{3} + 81 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 79 T^{2} - 320 T^{3} + 79 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 14 T + 149 T^{2} + 1104 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 113 T^{2} - 260 T^{3} + 113 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 231 T^{2} - 1776 T^{3} + 231 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 141 T^{2} + 132 T^{3} + 141 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 155 T^{2} + 1212 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 141 T^{2} + 736 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 197 T^{2} - 1384 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 3 p T^{2} - 1736 T^{3} + 3 p^{2} T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 269 T^{2} + 2400 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 30 T + 509 T^{2} + 5504 T^{3} + 509 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 531 T^{2} - 6040 T^{3} + 531 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.59697627056810919867425911908, −7.40137353331732613697697323450, −7.23052958930592378318656894902, −6.99794648745079350428838989429, −6.49567971302889179372872419451, −6.40881341252020935093690628261, −6.01752857913248424912033569996, −5.68945535701096759670973460187, −5.66634433935500030112312932891, −5.64457938647273418869038350734, −5.02363168094565849142993208315, −4.97811487647161179028334933791, −4.62421486453074319840843531418, −4.35350770596743812915312013949, −3.90226840973219944197339389158, −3.79544877020401997786366703596, −3.22728818203837251404544941036, −3.11044969479023232165022764089, −3.08900828706616228343418263650, −2.12314852338868049838079747333, −1.91758686782920236702047770151, −1.62501241086077114645374433798, −1.23641998370575971302954819884, −0.796265678733244511852275718911, −0.56364536844190097919749159071,
0.56364536844190097919749159071, 0.796265678733244511852275718911, 1.23641998370575971302954819884, 1.62501241086077114645374433798, 1.91758686782920236702047770151, 2.12314852338868049838079747333, 3.08900828706616228343418263650, 3.11044969479023232165022764089, 3.22728818203837251404544941036, 3.79544877020401997786366703596, 3.90226840973219944197339389158, 4.35350770596743812915312013949, 4.62421486453074319840843531418, 4.97811487647161179028334933791, 5.02363168094565849142993208315, 5.64457938647273418869038350734, 5.66634433935500030112312932891, 5.68945535701096759670973460187, 6.01752857913248424912033569996, 6.40881341252020935093690628261, 6.49567971302889179372872419451, 6.99794648745079350428838989429, 7.23052958930592378318656894902, 7.40137353331732613697697323450, 7.59697627056810919867425911908
