| L(s) = 1 | − 4·2-s − 6·3-s + 11·4-s + 24·6-s − 20·7-s − 18·8-s + 4·9-s + 4·11-s − 66·12-s + 80·14-s + 24·16-s − 30·17-s − 16·18-s − 10·19-s + 120·21-s − 16·22-s + 30·23-s + 108·24-s + 48·27-s − 220·28-s − 2·29-s − 20·31-s − 40·32-s − 24·33-s + 120·34-s + 44·36-s − 38·37-s + ⋯ |
| L(s) = 1 | − 2·2-s − 2·3-s + 11/4·4-s + 4·6-s − 2.85·7-s − 9/4·8-s + 4/9·9-s + 4/11·11-s − 5.5·12-s + 40/7·14-s + 3/2·16-s − 1.76·17-s − 8/9·18-s − 0.526·19-s + 40/7·21-s − 0.727·22-s + 1.30·23-s + 9/2·24-s + 16/9·27-s − 7.85·28-s − 0.0689·29-s − 0.645·31-s − 5/4·32-s − 0.727·33-s + 3.52·34-s + 11/9·36-s − 1.02·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07659520061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07659520061\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + 5 T^{2} - 3 p T^{3} - 31 T^{4} - 3 p^{3} T^{5} + 5 p^{4} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 p T + 32 T^{2} + 40 p T^{3} + 427 T^{4} + 40 p^{3} T^{5} + 32 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 20 T + 164 T^{2} + 636 T^{3} + 1871 T^{4} + 636 p^{2} T^{5} + 164 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 960 p^{2} T^{5} + 200 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 30 T + 109 T^{2} + 6390 T^{3} + 282060 T^{4} + 6390 p^{2} T^{5} + 109 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 1752 T^{3} - 96625 T^{4} + 1752 p^{2} T^{5} + 74 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30 T - 356 T^{2} - 5940 T^{3} + 821595 T^{4} - 5940 p^{2} T^{5} - 356 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 1571 T^{2} - 214 T^{3} + 1769980 T^{4} - 214 p^{2} T^{5} - 1571 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 18300 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 38 T + 41 p T^{2} - 52158 T^{3} - 1571608 T^{4} - 52158 p^{2} T^{5} + 41 p^{5} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 7272 p^{2} T^{5} + 3461 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 998 T^{2} - 2422797 T^{4} - 998 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} + 148716 T^{3} + 9565454 T^{4} + 148716 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4814 T^{2} + 12616659 T^{4} - 4814 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 164 T + 6980 T^{2} + 551844 T^{3} - 68910913 T^{4} + 551844 p^{2} T^{5} + 6980 p^{4} T^{6} - 164 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 312604 p^{2} T^{5} + 5413 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 170 T + 10706 T^{2} - 134496 T^{3} - 16049425 T^{4} - 134496 p^{2} T^{5} + 10706 p^{4} T^{6} - 170 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 258336 p^{2} T^{5} + 4658 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 58 T + 1682 T^{2} + 201144 T^{3} + 20590727 T^{4} + 201144 p^{2} T^{5} + 1682 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 7290 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 188 T + 17672 T^{2} + 1935084 T^{3} + 200304482 T^{4} + 1935084 p^{2} T^{5} + 17672 p^{4} T^{6} + 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 110 T + 12050 T^{2} - 1194180 T^{3} + 87746159 T^{4} - 1194180 p^{2} T^{5} + 12050 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 146 T + 13250 T^{2} - 1056876 T^{3} + 58176719 T^{4} - 1056876 p^{2} T^{5} + 13250 p^{4} T^{6} - 146 p^{6} T^{7} + p^{8} T^{8} \) |
| 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
−8.549824628092250002808571307592, −7.72163070755098720240028159261, −7.67914438918151809359547724814, −7.40240451087914594733210830670, −7.00347163290493059171398138781, −6.93348985549856683206472097540, −6.83638484408102715407299485046, −6.24477560848506090494094472306, −6.15913940874973328486458672184, −6.13270755573555773390916025755, −5.94069465167685243387582774730, −5.47174950563264775180291538562, −5.23429223550542158039676754075, −4.67215478694790483910844266146, −4.63675398556220252569553000472, −4.06366042064414817200699287689, −3.49399555131122687224699167091, −3.33524992264689316539636120649, −3.17081663817327375571820153797, −2.44267403398850754790402624771, −2.21553818563812951929833833454, −1.90079463055942759189088394013, −0.861243579343531291392818016906, −0.66923307631524300125628859900, −0.19922200987649567890267980118,
0.19922200987649567890267980118, 0.66923307631524300125628859900, 0.861243579343531291392818016906, 1.90079463055942759189088394013, 2.21553818563812951929833833454, 2.44267403398850754790402624771, 3.17081663817327375571820153797, 3.33524992264689316539636120649, 3.49399555131122687224699167091, 4.06366042064414817200699287689, 4.63675398556220252569553000472, 4.67215478694790483910844266146, 5.23429223550542158039676754075, 5.47174950563264775180291538562, 5.94069465167685243387582774730, 6.13270755573555773390916025755, 6.15913940874973328486458672184, 6.24477560848506090494094472306, 6.83638484408102715407299485046, 6.93348985549856683206472097540, 7.00347163290493059171398138781, 7.40240451087914594733210830670, 7.67914438918151809359547724814, 7.72163070755098720240028159261, 8.549824628092250002808571307592
