| L(s) = 1 | + 12·3-s + 12·5-s + 72·9-s + 44·11-s + 20·13-s + 144·15-s − 208·17-s + 116·19-s + 72·25-s + 132·27-s + 220·29-s − 176·31-s + 528·33-s − 348·37-s + 240·39-s − 44·43-s + 864·45-s + 720·47-s − 116·49-s − 2.49e3·51-s + 612·53-s + 528·55-s + 1.39e3·57-s + 1.79e3·59-s + 852·61-s + 240·65-s + 2.29e3·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.07·5-s + 8/3·9-s + 1.20·11-s + 0.426·13-s + 2.47·15-s − 2.96·17-s + 1.40·19-s + 0.575·25-s + 0.940·27-s + 1.40·29-s − 1.01·31-s + 2.78·33-s − 1.54·37-s + 0.985·39-s − 0.156·43-s + 2.86·45-s + 2.23·47-s − 0.338·49-s − 6.85·51-s + 1.58·53-s + 1.29·55-s + 3.23·57-s + 3.96·59-s + 1.78·61-s + 0.457·65-s + 4.17·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.92769054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.92769054\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 p T + 8 p^{2} T^{2} - 44 p T^{3} - 14 T^{4} - 44 p^{4} T^{5} + 8 p^{8} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 84 T^{3} - 13826 T^{4} - 84 p^{3} T^{5} + 72 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 116 T^{2} + 54790 T^{4} + 116 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 p T + 8 p^{2} T^{2} + 372 p T^{3} - 2010478 T^{4} + 372 p^{4} T^{5} + 8 p^{8} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 20460 T^{3} + 714782 T^{4} - 20460 p^{3} T^{5} + 200 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 104 T + 12258 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 116 T + 6728 T^{2} - 986812 T^{3} + 142022194 T^{4} - 986812 p^{3} T^{5} + 6728 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 404 p T^{2} + 314827206 T^{4} - 404 p^{7} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 220 T + 24200 T^{2} - 5230500 T^{3} + 1130124254 T^{4} - 5230500 p^{3} T^{5} + 24200 p^{6} T^{6} - 220 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 88 T + 48190 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 348 T + 60552 T^{2} + 14896836 T^{3} + 3603318782 T^{4} + 14896836 p^{3} T^{5} + 60552 p^{6} T^{6} + 348 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 186980 T^{2} + 17544268582 T^{4} - 186980 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 44 T + 968 T^{2} + 1233540 T^{3} - 1077444334 T^{4} + 1233540 p^{3} T^{5} + 968 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 360 T + 178846 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 612 T + 187272 T^{2} - 5820732 T^{3} - 19241963714 T^{4} - 5820732 p^{3} T^{5} + 187272 p^{6} T^{6} - 612 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1796 T + 1612808 T^{2} - 1066082252 T^{3} + 553985601874 T^{4} - 1066082252 p^{3} T^{5} + 1612808 p^{6} T^{6} - 1796 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 852 T + 362952 T^{2} - 267799788 T^{3} + 189964941278 T^{4} - 267799788 p^{3} T^{5} + 362952 p^{6} T^{6} - 852 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2292 T + 2626632 T^{2} - 2109545340 T^{3} + 1310310096626 T^{4} - 2109545340 p^{3} T^{5} + 2626632 p^{6} T^{6} - 2292 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1201676 T^{2} + 606649761286 T^{4} - 1201676 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 954684 T^{2} + 450988317542 T^{4} - 954684 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1600 T + 1469406 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1732 T + 1499912 T^{2} + 1638324780 T^{3} + 1649538614066 T^{4} + 1638324780 p^{3} T^{5} + 1499912 p^{6} T^{6} + 1732 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2607612 T^{2} + 2686461102758 T^{4} - 2607612 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 280 T + 1342018 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.41852465301355316623002627589, −7.24498401426239285767398315140, −6.88847110297224013371161514488, −6.80074665446038609647547553625, −6.70738425584075378703264809894, −6.36423654407895281202781632884, −6.01309069704626055929408782618, −5.56220125745344966524612995226, −5.34715794520567956413778526372, −5.27435289531188938400147612026, −5.12704646188832541549867590278, −4.29544143064264842970975626029, −4.07715588510139621631940523138, −4.06779039888721005837070829205, −3.80226014676093481579630777777, −3.73519172028901998995913515831, −2.92231078147261681282063969030, −2.73124718640472980739211247694, −2.69108585860721981894157313604, −2.29363139388491775177355755269, −1.97073487517536676920695749711, −1.68160433084345348961848029573, −1.37221211598364495919039588401, −0.842183515970383637969285713099, −0.42541610577512488633957169477,
0.42541610577512488633957169477, 0.842183515970383637969285713099, 1.37221211598364495919039588401, 1.68160433084345348961848029573, 1.97073487517536676920695749711, 2.29363139388491775177355755269, 2.69108585860721981894157313604, 2.73124718640472980739211247694, 2.92231078147261681282063969030, 3.73519172028901998995913515831, 3.80226014676093481579630777777, 4.06779039888721005837070829205, 4.07715588510139621631940523138, 4.29544143064264842970975626029, 5.12704646188832541549867590278, 5.27435289531188938400147612026, 5.34715794520567956413778526372, 5.56220125745344966524612995226, 6.01309069704626055929408782618, 6.36423654407895281202781632884, 6.70738425584075378703264809894, 6.80074665446038609647547553625, 6.88847110297224013371161514488, 7.24498401426239285767398315140, 7.41852465301355316623002627589
