| L(s) = 1 | − 18·3-s + 53·5-s − 6·7-s + 81·9-s + 191·11-s − 758·13-s − 954·15-s + 340·17-s − 1.76e3·19-s + 108·21-s + 3.23e3·23-s + 4.55e3·25-s + 1.45e3·27-s + 8.91e3·29-s + 1.99e3·31-s − 3.43e3·33-s − 318·35-s − 2.05e4·37-s + 1.36e4·39-s + 1.76e4·41-s − 3.17e4·43-s + 4.29e3·45-s + 3.39e4·47-s + 4.81e3·49-s − 6.12e3·51-s − 4.92e4·53-s + 1.01e4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.948·5-s − 0.0462·7-s + 1/3·9-s + 0.475·11-s − 1.24·13-s − 1.09·15-s + 0.285·17-s − 1.12·19-s + 0.0534·21-s + 1.27·23-s + 1.45·25-s + 0.384·27-s + 1.96·29-s + 0.372·31-s − 0.549·33-s − 0.0438·35-s − 2.47·37-s + 1.43·39-s + 1.63·41-s − 2.61·43-s + 0.316·45-s + 2.23·47-s + 0.286·49-s − 0.329·51-s − 2.40·53-s + 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.128169879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128169879\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T - 683 p T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 53 T - 1743 T^{2} + 89994 T^{3} + 2176954 T^{4} + 89994 p^{5} T^{5} - 1743 p^{10} T^{6} - 53 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 191 T - 234735 T^{2} + 883566 p T^{3} + 345003244 p^{2} T^{4} + 883566 p^{6} T^{5} - 234735 p^{10} T^{6} - 191 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 379 T + 488066 T^{2} + 379 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 p T - 1792914 T^{2} + 18624000 p T^{3} + 1462296318547 T^{4} + 18624000 p^{6} T^{5} - 1792914 p^{10} T^{6} - 20 p^{16} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 1769 T + 1045603 T^{2} - 5074270360 T^{3} - 9537626497976 T^{4} - 5074270360 p^{5} T^{5} + 1045603 p^{10} T^{6} + 1769 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3236 T - 4980510 T^{2} - 8347326720 T^{3} + 129944731805059 T^{4} - 8347326720 p^{5} T^{5} - 4980510 p^{10} T^{6} - 3236 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4459 T + 37061638 T^{2} - 4459 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1994 T + 10280849 T^{2} + 126744851310 T^{3} - 893708155041268 T^{4} + 126744851310 p^{5} T^{5} + 10280849 p^{10} T^{6} - 1994 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20587 T + 185423563 T^{2} + 2052793425004 T^{3} + 22636782601114654 T^{4} + 2052793425004 p^{5} T^{5} + 185423563 p^{10} T^{6} + 20587 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8814 T + 240678562 T^{2} - 8814 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 15853 T + 338678796 T^{2} + 15853 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 33912 T + 462240278 T^{2} - 7769017144224 T^{3} + 156694760249296899 T^{4} - 7769017144224 p^{5} T^{5} + 462240278 p^{10} T^{6} - 33912 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 49239 T + 1103481011 T^{2} + 23861570178636 T^{3} + 556242294983257398 T^{4} + 23861570178636 p^{5} T^{5} + 1103481011 p^{10} T^{6} + 49239 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 56735 T + 988331391 T^{2} + 45426593189460 T^{3} + 2162900736067650880 T^{4} + 45426593189460 p^{5} T^{5} + 988331391 p^{10} T^{6} + 56735 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 67508 T + 1731262802 T^{2} + 76748134547280 T^{3} + 3424209143194800779 T^{4} + 76748134547280 p^{5} T^{5} + 1731262802 p^{10} T^{6} + 67508 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 75723 T + 1872823325 T^{2} - 87906769364370 T^{3} + 5344056371181586764 T^{4} - 87906769364370 p^{5} T^{5} + 1872823325 p^{10} T^{6} - 75723 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8992 T - 681216182 T^{2} - 8992 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3201 T - 2300766979 T^{2} + 5874250509006 T^{3} + 1021915489991879958 T^{4} + 5874250509006 p^{5} T^{5} - 2300766979 p^{10} T^{6} - 3201 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 26612 T - 3289450441 T^{2} + 57387814991556 T^{3} + 4333754463066999968 T^{4} + 57387814991556 p^{5} T^{5} - 3289450441 p^{10} T^{6} - 26612 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 949 T + 367057696 T^{2} - 949 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 176562 T + 12318337010 T^{2} + 1357352851108032 T^{3} + \)\(15\!\cdots\!99\)\( T^{4} + 1357352851108032 p^{5} T^{5} + 12318337010 p^{10} T^{6} + 176562 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 129423 T + 11942811256 T^{2} + 129423 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.43405440489018828499527026641, −7.17049658544612904984236278780, −6.84376097425132941238562797928, −6.69144271671169524909719445782, −6.58710765598126271228871293941, −6.08977406802697346458534352411, −6.00202521184284680332592697544, −5.90946677805727893461049424890, −5.16599322793603760425774586350, −5.16161245833028639427157226494, −5.08141941541357539426494216769, −4.64945514098815285281246171749, −4.42445899417816620148550033226, −4.30010729937356857222730498208, −3.55264311809499037253193918691, −3.39954939959838367557642014922, −2.91126666527624320595977150986, −2.80425232209686859272645099190, −2.45068996262327408558974267907, −2.00646339001160904213629184629, −1.53371413056489809555284023335, −1.39998275367882146143241893666, −0.974320334314310470347957917689, −0.52552397027707958956902202247, −0.17772440358612500032284825227,
0.17772440358612500032284825227, 0.52552397027707958956902202247, 0.974320334314310470347957917689, 1.39998275367882146143241893666, 1.53371413056489809555284023335, 2.00646339001160904213629184629, 2.45068996262327408558974267907, 2.80425232209686859272645099190, 2.91126666527624320595977150986, 3.39954939959838367557642014922, 3.55264311809499037253193918691, 4.30010729937356857222730498208, 4.42445899417816620148550033226, 4.64945514098815285281246171749, 5.08141941541357539426494216769, 5.16161245833028639427157226494, 5.16599322793603760425774586350, 5.90946677805727893461049424890, 6.00202521184284680332592697544, 6.08977406802697346458534352411, 6.58710765598126271228871293941, 6.69144271671169524909719445782, 6.84376097425132941238562797928, 7.17049658544612904984236278780, 7.43405440489018828499527026641
