| L(s) = 1 | + 3·2-s − 18·3-s + 18·4-s + 72·5-s − 54·6-s − 57·8-s + 81·9-s + 216·10-s − 480·11-s − 324·12-s − 2.59e3·13-s − 1.29e3·15-s + 565·16-s + 936·17-s + 243·18-s − 216·19-s + 1.29e3·20-s − 1.44e3·22-s + 504·23-s + 1.02e3·24-s + 7.54e3·25-s − 7.77e3·26-s + 1.45e3·27-s + 1.27e4·29-s − 3.88e3·30-s + 9.93e3·31-s − 2.75e3·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 1.15·3-s + 9/16·4-s + 1.28·5-s − 0.612·6-s − 0.314·8-s + 1/3·9-s + 0.683·10-s − 1.19·11-s − 0.649·12-s − 4.25·13-s − 1.48·15-s + 0.551·16-s + 0.785·17-s + 0.176·18-s − 0.137·19-s + 0.724·20-s − 0.634·22-s + 0.198·23-s + 0.363·24-s + 2.41·25-s − 2.25·26-s + 0.384·27-s + 2.81·29-s − 0.788·30-s + 1.85·31-s − 0.475·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.671632785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671632785\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T - 9 T^{2} + 69 p T^{3} - 247 p^{2} T^{4} + 69 p^{6} T^{5} - 9 p^{10} T^{6} - 3 p^{15} T^{7} + p^{20} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 36 T - 1829 T^{2} - 36 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 480 T - 146214 T^{2} + 26165760 T^{3} + 78794529995 T^{4} + 26165760 p^{5} T^{5} - 146214 p^{10} T^{6} + 480 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 1296 T + 912362 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 936 T + 4030 p T^{2} + 1902071808 T^{3} - 2607729791325 T^{4} + 1902071808 p^{5} T^{5} + 4030 p^{11} T^{6} - 936 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 216 T - 915158 T^{2} - 861922944 T^{3} - 5321504962149 T^{4} - 861922944 p^{5} T^{5} - 915158 p^{10} T^{6} + 216 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 504 T - 12530862 T^{2} + 44255232 T^{3} + 120391660157747 T^{4} + 44255232 p^{5} T^{5} - 12530862 p^{10} T^{6} - 504 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6372 T + 31409694 T^{2} - 6372 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 9936 T + 32792962 T^{2} - 86173258752 T^{3} + 700899091822659 T^{4} - 86173258752 p^{5} T^{5} + 32792962 p^{10} T^{6} - 9936 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 11124 T + 7924930 T^{2} - 254399962032 T^{3} + 24472717058235 T^{4} - 254399962032 p^{5} T^{5} + 7924930 p^{10} T^{6} + 11124 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 20952 T + 339207826 T^{2} + 20952 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 6264 T + 25906310 T^{2} + 6264 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7920 T + 164649698 T^{2} + 4440057431040 T^{3} - 56596760207718045 T^{4} + 4440057431040 p^{5} T^{5} + 164649698 p^{10} T^{6} - 7920 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2220 T - 765055134 T^{2} - 147424543440 T^{3} + 415926137597914907 T^{4} - 147424543440 p^{5} T^{5} - 765055134 p^{10} T^{6} + 2220 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 504 p T - 730657894 T^{2} - 93259651968 p T^{3} + 1464587879467583595 T^{4} - 93259651968 p^{6} T^{5} - 730657894 p^{10} T^{6} - 504 p^{16} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 17280 T - 408453002 T^{2} - 16971399936000 T^{3} - 465616330717235397 T^{4} - 16971399936000 p^{5} T^{5} - 408453002 p^{10} T^{6} + 17280 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20680 T - 353466614 T^{2} + 592349648000 p T^{3} - 398768770683677 p^{2} T^{4} + 592349648000 p^{6} T^{5} - 353466614 p^{10} T^{6} - 20680 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 92280 T + 5423120334 T^{2} + 92280 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 56592 T - 1575065810 T^{2} - 35742210564096 T^{3} + 11889653195374036275 T^{4} - 35742210564096 p^{5} T^{5} - 1575065810 p^{10} T^{6} - 56592 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 56096 T - 1768005086 T^{2} + 69522381039616 T^{3} + 5003889984830086819 T^{4} + 69522381039616 p^{5} T^{5} - 1768005086 p^{10} T^{6} - 56096 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 71352 T + 4792627990 T^{2} - 71352 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 123192 T + 864665678 T^{2} - 387252116407296 T^{3} + 88895078272975471059 T^{4} - 387252116407296 p^{5} T^{5} + 864665678 p^{10} T^{6} - 123192 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 35856 T - 1009376254 T^{2} + 35856 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.404747180072119851763403820929, −8.377655474402587694959674414487, −8.206762316558817103292743599757, −7.65275987164542417806405588247, −7.18036492138922651352710120275, −7.08836076275241121556252795694, −7.01453773230525388392603111734, −6.46792368311945018344141513372, −6.26971477684641125710902287419, −6.15056555912752834632810224665, −5.40684809445614678000790239515, −5.16396460015153211399873043696, −5.13393218386062056853210059318, −5.08375222837157653917740167971, −4.64426629175017178932351141628, −4.45861276921992137466740446258, −3.38238524204712429410322952491, −3.08689152358515393756691772641, −2.94701482287230663160190025542, −2.57759338443716562144763363260, −2.18324051894513451154729981862, −1.84892524201049896795081606936, −1.19366849530245403822619531518, −0.49244682847355574768995362604, −0.38972979076934287246565437279,
0.38972979076934287246565437279, 0.49244682847355574768995362604, 1.19366849530245403822619531518, 1.84892524201049896795081606936, 2.18324051894513451154729981862, 2.57759338443716562144763363260, 2.94701482287230663160190025542, 3.08689152358515393756691772641, 3.38238524204712429410322952491, 4.45861276921992137466740446258, 4.64426629175017178932351141628, 5.08375222837157653917740167971, 5.13393218386062056853210059318, 5.16396460015153211399873043696, 5.40684809445614678000790239515, 6.15056555912752834632810224665, 6.26971477684641125710902287419, 6.46792368311945018344141513372, 7.01453773230525388392603111734, 7.08836076275241121556252795694, 7.18036492138922651352710120275, 7.65275987164542417806405588247, 8.206762316558817103292743599757, 8.377655474402587694959674414487, 8.404747180072119851763403820929
