| L(s) = 1 | + 18·5-s − 2·7-s + 18·11-s − 10·13-s + 40·19-s + 18·23-s + 139·25-s − 18·29-s − 38·31-s − 36·35-s + 128·37-s + 126·41-s + 46·43-s + 54·47-s + 45·49-s + 324·55-s + 126·59-s + 62·61-s − 180·65-s + 106·67-s − 208·73-s − 36·77-s − 14·79-s − 378·83-s + 20·91-s + 720·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 18/5·5-s − 2/7·7-s + 1.63·11-s − 0.769·13-s + 2.10·19-s + 0.782·23-s + 5.55·25-s − 0.620·29-s − 1.22·31-s − 1.02·35-s + 3.45·37-s + 3.07·41-s + 1.06·43-s + 1.14·47-s + 0.918·49-s + 5.89·55-s + 2.13·59-s + 1.01·61-s − 2.76·65-s + 1.58·67-s − 2.84·73-s − 0.467·77-s − 0.177·79-s − 4.55·83-s + 0.219·91-s + 7.57·95-s + 0.144·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.63315434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.63315434\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 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
−7.889780161252355754385042960924, −7.30935743803192960465605270216, −7.21839078051065151387208775637, −7.14042856532831997281312196859, −7.05402285207911622031215150336, −6.38455822614991526323218099808, −6.17900990589495785548604398217, −5.95008617175208550704634382119, −5.91035756359968016568483141297, −5.69339449736989834912973326555, −5.48923280818245499252296549719, −5.15656228100934161487730936077, −5.02259582802466201235199990554, −4.30828407233185125079706004098, −4.20098954781521286550830100874, −3.93986524255537947830211132552, −3.69409856005462168381066599436, −2.74822599179043403301668912659, −2.73341082197045377242081591206, −2.66132126227914873931351444642, −2.28633801740749565461564413357, −1.70241209274272067272022356836, −1.47805463494435700163897588358, −0.916666247310705112355900481396, −0.876060881522653085421041902336,
0.876060881522653085421041902336, 0.916666247310705112355900481396, 1.47805463494435700163897588358, 1.70241209274272067272022356836, 2.28633801740749565461564413357, 2.66132126227914873931351444642, 2.73341082197045377242081591206, 2.74822599179043403301668912659, 3.69409856005462168381066599436, 3.93986524255537947830211132552, 4.20098954781521286550830100874, 4.30828407233185125079706004098, 5.02259582802466201235199990554, 5.15656228100934161487730936077, 5.48923280818245499252296549719, 5.69339449736989834912973326555, 5.91035756359968016568483141297, 5.95008617175208550704634382119, 6.17900990589495785548604398217, 6.38455822614991526323218099808, 7.05402285207911622031215150336, 7.14042856532831997281312196859, 7.21839078051065151387208775637, 7.30935743803192960465605270216, 7.889780161252355754385042960924
