| L(s) = 1 | − 4·5-s + 6·7-s − 52·11-s − 26·13-s + 376·17-s − 148·19-s − 148·23-s + 74·25-s − 288·29-s + 248·31-s − 24·35-s + 684·37-s + 256·43-s − 132·47-s − 25·49-s + 1.90e3·53-s + 208·55-s + 1.00e3·59-s + 34·61-s + 104·65-s + 866·67-s + 1.55e3·71-s + 3.74e3·73-s − 312·77-s − 182·79-s + 1.33e3·83-s − 1.50e3·85-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 0.323·7-s − 1.42·11-s − 0.554·13-s + 5.36·17-s − 1.78·19-s − 1.34·23-s + 0.591·25-s − 1.84·29-s + 1.43·31-s − 0.115·35-s + 3.03·37-s + 0.907·43-s − 0.409·47-s − 0.0728·49-s + 4.93·53-s + 0.509·55-s + 2.21·59-s + 0.0713·61-s + 0.198·65-s + 1.57·67-s + 2.59·71-s + 6.00·73-s − 0.461·77-s − 0.259·79-s + 1.76·83-s − 1.91·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.613639818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.613639818\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T - 58 T^{2} - 704 T^{3} - 12149 T^{4} - 704 p^{3} T^{5} - 58 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 61 T^{2} + 4266 T^{3} - 129372 T^{4} + 4266 p^{3} T^{5} + 61 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 52 T + 986 T^{2} - 49088 T^{3} - 2419061 T^{4} - 49088 p^{3} T^{5} + 986 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 p T - 3167 T^{2} - 1102 p T^{3} + 8456668 T^{4} - 1102 p^{4} T^{5} - 3167 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 188 T + 18482 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 74 T + 3567 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 148 T - 3406 T^{2} + 144448 T^{3} + 226054243 T^{4} + 144448 p^{3} T^{5} - 3406 p^{6} T^{6} + 148 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 288 T + 24950 T^{2} + 2654208 T^{3} + 745559499 T^{4} + 2654208 p^{3} T^{5} + 24950 p^{6} T^{6} + 288 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 p T - 1934 T^{2} - 30848 p T^{3} + 1304610499 T^{4} - 30848 p^{4} T^{5} - 1934 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 342 T + 112547 T^{2} - 342 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 46478 T^{2} - 2589899757 T^{4} + 46478 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 256 T - 106982 T^{2} - 3457024 T^{3} + 18230526523 T^{4} - 3457024 p^{3} T^{5} - 106982 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132 T - 194398 T^{2} + 551232 T^{3} + 32280332403 T^{4} + 551232 p^{3} T^{5} - 194398 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 952 T + 489050 T^{2} - 952 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1004 T + 424634 T^{2} - 173314496 T^{3} + 91128706219 T^{4} - 173314496 p^{3} T^{5} + 424634 p^{6} T^{6} - 1004 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 34 T - 245015 T^{2} + 7064894 T^{3} + 8817396844 T^{4} + 7064894 p^{3} T^{5} - 245015 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 866 T + 7021 T^{2} - 122460194 T^{3} + 235935015628 T^{4} - 122460194 p^{3} T^{5} + 7021 p^{6} T^{6} - 866 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 776 T + 704366 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 182 T - 839555 T^{2} - 20638618 T^{3} + 502149757684 T^{4} - 20638618 p^{3} T^{5} - 839555 p^{6} T^{6} + 182 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1336 T + 575978 T^{2} - 87299584 T^{3} + 113962028203 T^{4} - 87299584 p^{3} T^{5} + 575978 p^{6} T^{6} - 1336 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 876 T + 1522402 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 38 T - 430343 T^{2} + 52955242 T^{3} - 647849891372 T^{4} + 52955242 p^{3} T^{5} - 430343 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.32966936740005056693696897375, −7.08061732797020980394908811383, −6.67627491874674725461919660044, −6.45227540643231598705175701565, −6.18891970547830072934430100841, −5.80469989618407623073892996550, −5.69788111212983195763211504583, −5.49724037985137457570934949406, −5.38468503701235760481462720872, −5.01095868415555860512140546012, −4.78144034109563403202799788473, −4.62775463683312949195832914169, −4.01817254202835137098100178125, −3.72299854824730308039938188649, −3.68680962001030151557493391691, −3.49907683495338281909128341131, −3.27311616708989915692539026008, −2.36759177346726784238267057638, −2.34842932744882721930734083325, −2.33543601761627608370671790317, −2.14474416918060711779626530175, −1.02569938515430774526045928938, −0.893005918541095236221591117660, −0.793626959977809006047739583763, −0.53128270204561195237311779156,
0.53128270204561195237311779156, 0.793626959977809006047739583763, 0.893005918541095236221591117660, 1.02569938515430774526045928938, 2.14474416918060711779626530175, 2.33543601761627608370671790317, 2.34842932744882721930734083325, 2.36759177346726784238267057638, 3.27311616708989915692539026008, 3.49907683495338281909128341131, 3.68680962001030151557493391691, 3.72299854824730308039938188649, 4.01817254202835137098100178125, 4.62775463683312949195832914169, 4.78144034109563403202799788473, 5.01095868415555860512140546012, 5.38468503701235760481462720872, 5.49724037985137457570934949406, 5.69788111212983195763211504583, 5.80469989618407623073892996550, 6.18891970547830072934430100841, 6.45227540643231598705175701565, 6.67627491874674725461919660044, 7.08061732797020980394908811383, 7.32966936740005056693696897375
