| L(s) = 1 | + 12·7-s − 16·11-s − 48·13-s + 24·17-s + 56·23-s − 80·31-s − 112·37-s − 56·41-s + 8·43-s + 16·47-s + 72·49-s + 120·53-s − 24·61-s + 8·67-s + 272·71-s − 108·73-s − 192·77-s − 9·81-s − 272·83-s − 576·91-s + 148·97-s + 152·101-s − 124·103-s + 160·107-s + 144·113-s + 288·119-s − 312·121-s + ⋯ |
| L(s) = 1 | + 12/7·7-s − 1.45·11-s − 3.69·13-s + 1.41·17-s + 2.43·23-s − 2.58·31-s − 3.02·37-s − 1.36·41-s + 8/43·43-s + 0.340·47-s + 1.46·49-s + 2.26·53-s − 0.393·61-s + 8/67·67-s + 3.83·71-s − 1.47·73-s − 2.49·77-s − 1/9·81-s − 3.27·83-s − 6.32·91-s + 1.52·97-s + 1.50·101-s − 1.20·103-s + 1.49·107-s + 1.27·113-s + 2.42·119-s − 2.57·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7625000791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7625000791\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6014 T^{4} - 660 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 252 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 21360 T^{3} + 319874 T^{4} + 21360 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} + 5448 T^{3} - 163198 T^{4} + 5448 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1004 T^{2} + 509190 T^{4} - 1004 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 45528 T^{3} + 1241282 T^{4} - 45528 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1424 T^{2} + 980610 T^{4} - 1424 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 T + 1722 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 316848 T^{3} + 13874882 T^{4} + 316848 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 2382 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 3960 T^{3} - 5004286 T^{4} + 3960 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} + 33456 T^{3} - 9745438 T^{4} + 33456 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 409080 T^{3} + 22882562 T^{4} - 409080 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8600 T^{2} + 42555378 T^{4} - 8600 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 4574 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 75000 T^{3} - 16429246 T^{4} + 75000 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 136 T + 14322 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 108 T + 5832 T^{2} + 214596 T^{3} - 3272626 T^{4} + 214596 p^{2} T^{5} + 5832 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 556 T^{2} + 67139430 T^{4} + 556 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 272 T + 36992 T^{2} + 3994320 T^{3} + 370520834 T^{4} + 3994320 p^{2} T^{5} + 36992 p^{4} T^{6} + 272 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23804 T^{2} + 265665990 T^{4} - 23804 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 74 T + 2738 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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
−7.30806514086696473301166583373, −7.23208962093387218638124238322, −7.17893248459610444680036263502, −7.10671589831243265174638120665, −6.75455325581249492945517704185, −6.16074673440878731787531504755, −5.99802802620394000440673620203, −5.37699738158022334089270678293, −5.27387325740037518438428894674, −5.20972829156717111691192639855, −5.14662998741369115743340052170, −4.99176131211493128421701594453, −4.76075795104080574640807695651, −4.26085071725586201800214401220, −3.92417834099491537321120827732, −3.49850674645614249570349280038, −3.45376192085711554411419580138, −2.88163651706958315832135489688, −2.52469675522747206476936201636, −2.51128942930985888177676149284, −2.09323463057183216184145326516, −1.57768307882658398817181535380, −1.49653771265173974823756708096, −0.70283963971282750097145809492, −0.17545688932456435993932587014,
0.17545688932456435993932587014, 0.70283963971282750097145809492, 1.49653771265173974823756708096, 1.57768307882658398817181535380, 2.09323463057183216184145326516, 2.51128942930985888177676149284, 2.52469675522747206476936201636, 2.88163651706958315832135489688, 3.45376192085711554411419580138, 3.49850674645614249570349280038, 3.92417834099491537321120827732, 4.26085071725586201800214401220, 4.76075795104080574640807695651, 4.99176131211493128421701594453, 5.14662998741369115743340052170, 5.20972829156717111691192639855, 5.27387325740037518438428894674, 5.37699738158022334089270678293, 5.99802802620394000440673620203, 6.16074673440878731787531504755, 6.75455325581249492945517704185, 7.10671589831243265174638120665, 7.17893248459610444680036263502, 7.23208962093387218638124238322, 7.30806514086696473301166583373
