| L(s) = 1 | + 2·4-s − 2·7-s − 18·11-s + 10·13-s − 40·19-s + 18·23-s − 4·28-s − 18·29-s + 38·31-s − 128·37-s + 126·41-s + 46·43-s − 36·44-s + 54·47-s + 45·49-s + 20·52-s − 126·59-s + 62·61-s − 8·64-s + 106·67-s + 208·73-s − 80·76-s + 36·77-s + 14·79-s − 378·83-s − 20·91-s + 36·92-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/7·7-s − 1.63·11-s + 0.769·13-s − 2.10·19-s + 0.782·23-s − 1/7·28-s − 0.620·29-s + 1.22·31-s − 3.45·37-s + 3.07·41-s + 1.06·43-s − 0.818·44-s + 1.14·47-s + 0.918·49-s + 5/13·52-s − 2.13·59-s + 1.01·61-s − 1/8·64-s + 1.58·67-s + 2.84·73-s − 1.05·76-s + 0.467·77-s + 0.177·79-s − 4.55·83-s − 0.219·91-s + 9/23·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6936081138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6936081138\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 2 T - 41 T^{2} - 106 T^{3} - 572 T^{4} - 106 p^{2} T^{5} - 41 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 359 T^{2} + 4518 T^{3} + 61428 T^{4} + 4518 p^{2} T^{5} + 359 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 10 T - 47 T^{2} + 1910 T^{3} - 23852 T^{4} + 1910 p^{2} T^{5} - 47 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T + 606 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 18 T + 1175 T^{2} - 19206 T^{3} + 915780 T^{4} - 19206 p^{2} T^{5} + 1175 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 1745 T^{2} + 29466 T^{3} + 2063316 T^{4} + 29466 p^{2} T^{5} + 1745 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4750 p^{2} T^{5} - 353 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 64 T + 3546 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 126 T + 9329 T^{2} - 508662 T^{3} + 22367460 T^{4} - 508662 p^{2} T^{5} + 9329 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 46 T - 1625 T^{2} - 46 p T^{3} + 3604 p^{2} T^{4} - 46 p^{3} T^{5} - 1625 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 54 T + 4751 T^{2} - 204066 T^{3} + 11548308 T^{4} - 204066 p^{2} T^{5} + 4751 p^{4} T^{6} - 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 126 T + 10535 T^{2} + 660618 T^{3} + 33793140 T^{4} + 660618 p^{2} T^{5} + 10535 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 60946 p^{2} T^{5} - 2615 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 106 T - 65 T^{2} - 246238 T^{3} + 57123076 T^{4} - 246238 p^{2} T^{5} - 65 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 104 T + 11418 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 14 T - 10985 T^{2} + 18214 T^{3} + 84841444 T^{4} + 18214 p^{2} T^{5} - 10985 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 14 T - 8087 T^{2} - 147490 T^{3} - 21765356 T^{4} - 147490 p^{2} T^{5} - 8087 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.63816776345113413259237284051, −6.44007419443045506936878320291, −6.23192386995540756924170101135, −6.06898463392775754406595718450, −5.72503869738489734421989894640, −5.54842572764244440369432192178, −5.41544274881062633448873351268, −5.16883698739982193455215484777, −4.98780018023317053698921620346, −4.62051616412171688499763291451, −4.22042085350499742619584114496, −4.08801361869768317593984850740, −4.04680633059770027971822453404, −3.86101694537025759778716855710, −3.14962759775545381102083836155, −3.08936336533314765301193820500, −2.96026422429486332351215017785, −2.58379186360018931995455259436, −2.21120404266511714670315562970, −2.16883423699024416277877747231, −1.83490227054982953249005024787, −1.42289655386672240879167012473, −0.816808833154251492841654050464, −0.76574576585069612822132675272, −0.12129846087631780215650361383,
0.12129846087631780215650361383, 0.76574576585069612822132675272, 0.816808833154251492841654050464, 1.42289655386672240879167012473, 1.83490227054982953249005024787, 2.16883423699024416277877747231, 2.21120404266511714670315562970, 2.58379186360018931995455259436, 2.96026422429486332351215017785, 3.08936336533314765301193820500, 3.14962759775545381102083836155, 3.86101694537025759778716855710, 4.04680633059770027971822453404, 4.08801361869768317593984850740, 4.22042085350499742619584114496, 4.62051616412171688499763291451, 4.98780018023317053698921620346, 5.16883698739982193455215484777, 5.41544274881062633448873351268, 5.54842572764244440369432192178, 5.72503869738489734421989894640, 6.06898463392775754406595718450, 6.23192386995540756924170101135, 6.44007419443045506936878320291, 6.63816776345113413259237284051
