| L(s) = 1 | + 4·2-s + 10·4-s + 4·5-s − 5·7-s + 20·8-s + 16·10-s − 9·11-s − 12·13-s − 20·14-s + 35·16-s + 4·19-s + 40·20-s − 36·22-s − 6·23-s + 10·25-s − 48·26-s − 50·28-s − 18·29-s + 2·31-s + 56·32-s − 20·35-s + 9·37-s + 16·38-s + 80·40-s − 12·41-s − 16·43-s − 90·44-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.78·5-s − 1.88·7-s + 7.07·8-s + 5.05·10-s − 2.71·11-s − 3.32·13-s − 5.34·14-s + 35/4·16-s + 0.917·19-s + 8.94·20-s − 7.67·22-s − 1.25·23-s + 2·25-s − 9.41·26-s − 9.44·28-s − 3.34·29-s + 0.359·31-s + 9.89·32-s − 3.38·35-s + 1.47·37-s + 2.59·38-s + 12.6·40-s − 1.87·41-s − 2.43·43-s − 13.5·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 5 T + 19 T^{2} + 30 T^{3} + 80 T^{4} + 30 p T^{5} + 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 9 T + 47 T^{2} + 212 T^{3} + 800 T^{4} + 212 p T^{5} + 47 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 12 T + 98 T^{2} + 526 T^{3} + 2214 T^{4} + 526 p T^{5} + 98 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 24 T^{2} + 98 T^{3} + 290 T^{4} + 98 p T^{5} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T - 4 T^{2} - 68 T^{3} + 790 T^{4} - 68 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 6 T + 48 T^{2} + 10 p T^{3} + 1118 T^{4} + 10 p^{2} T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 18 T + 206 T^{2} + 1600 T^{3} + 9854 T^{4} + 1600 p T^{5} + 206 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 2 T + 82 T^{2} - 240 T^{3} + 3146 T^{4} - 240 p T^{5} + 82 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 9 T + 107 T^{2} - 670 T^{3} + 5406 T^{4} - 670 p T^{5} + 107 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 12 T + 124 T^{2} + 702 T^{3} + 4810 T^{4} + 702 p T^{5} + 124 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 16 T + 236 T^{2} + 2034 T^{3} + 16350 T^{4} + 2034 p T^{5} + 236 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 10 T + 78 T^{2} + 612 T^{3} + 5978 T^{4} + 612 p T^{5} + 78 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 8 T + 102 T^{2} + 810 T^{3} + 8870 T^{4} + 810 p T^{5} + 102 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 18 T + 200 T^{2} + 1280 T^{3} + 8822 T^{4} + 1280 p T^{5} + 200 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 11 T + 139 T^{2} + 918 T^{3} + 9230 T^{4} + 918 p T^{5} + 139 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 27 T + 405 T^{2} + 4664 T^{3} + 44204 T^{4} + 4664 p T^{5} + 405 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 208 T^{2} - 458 T^{3} + 19258 T^{4} - 458 p T^{5} + 208 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 20 T + 380 T^{2} + 4388 T^{3} + 47814 T^{4} + 4388 p T^{5} + 380 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 9 T + 223 T^{2} + 2340 T^{3} + 23724 T^{4} + 2340 p T^{5} + 223 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 17 T + 437 T^{2} + 4606 T^{3} + 61882 T^{4} + 4606 p T^{5} + 437 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5 T + 277 T^{2} + 1126 T^{3} + 35698 T^{4} + 1126 p T^{5} + 277 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−5.83363030482663038585303706523, −5.76867255463411844489700492188, −5.63543218910766186215718660334, −5.42512735562724118643482561838, −5.39567178590668755462113157103, −5.03100261315792589491438842059, −4.96265084300298148836987006435, −4.83970137839385621247396092288, −4.72030185253288735008404869598, −4.53403320190455318121619761191, −4.06082509188680725578203844761, −3.96855445428470457212533417944, −3.90217474829089539333573267279, −3.28462293911345435658824895061, −3.28201003686040537821180867618, −3.04061380106911955873032280787, −3.01486404727421894627863555182, −2.70291427526641067191021739592, −2.55038294750255942841985839713, −2.52473292293264409599335904513, −2.31820173610210254847794851475, −1.78133942098203999076723859014, −1.54176215542456282095526764111, −1.53150345621710793163126109345, −1.52506966879871582148993395178, 0, 0, 0, 0,
1.52506966879871582148993395178, 1.53150345621710793163126109345, 1.54176215542456282095526764111, 1.78133942098203999076723859014, 2.31820173610210254847794851475, 2.52473292293264409599335904513, 2.55038294750255942841985839713, 2.70291427526641067191021739592, 3.01486404727421894627863555182, 3.04061380106911955873032280787, 3.28201003686040537821180867618, 3.28462293911345435658824895061, 3.90217474829089539333573267279, 3.96855445428470457212533417944, 4.06082509188680725578203844761, 4.53403320190455318121619761191, 4.72030185253288735008404869598, 4.83970137839385621247396092288, 4.96265084300298148836987006435, 5.03100261315792589491438842059, 5.39567178590668755462113157103, 5.42512735562724118643482561838, 5.63543218910766186215718660334, 5.76867255463411844489700492188, 5.83363030482663038585303706523
