| L(s) = 1 | + 2-s + 3·3-s − 4-s + 3·6-s + 4·7-s + 2·8-s − 9-s − 2·11-s − 3·12-s + 7·13-s + 4·14-s + 3·16-s + 17-s − 18-s + 12·21-s − 2·22-s − 2·23-s + 6·24-s + 7·26-s − 11·27-s − 4·28-s + 29-s − 2·32-s − 6·33-s + 34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s + 1.51·7-s + 0.707·8-s − 1/3·9-s − 0.603·11-s − 0.866·12-s + 1.94·13-s + 1.06·14-s + 3/4·16-s + 0.242·17-s − 0.235·18-s + 2.61·21-s − 0.426·22-s − 0.417·23-s + 1.22·24-s + 1.37·26-s − 2.11·27-s − 0.755·28-s + 0.185·29-s − 0.353·32-s − 1.04·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.39647906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.39647906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + p T^{2} - 5 T^{3} + 3 p T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - p T + 10 T^{2} - 22 T^{3} + 44 T^{4} - 22 p T^{5} + 10 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4 T + 27 T^{2} - 69 T^{3} + 272 T^{4} - 69 p T^{5} + 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 59 T^{2} - 250 T^{3} + 1180 T^{4} - 250 p T^{5} + 59 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - T + 38 T^{2} - 4 p T^{3} + 822 T^{4} - 4 p^{2} T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 75 T^{2} + 145 T^{3} + 2398 T^{4} + 145 p T^{5} + 75 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 53 T^{2} - 281 T^{3} + 1251 T^{4} - 281 p T^{5} + 53 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 57 T^{2} + 5 T^{3} + 2675 T^{4} + 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 2 T + 117 T^{2} + 99 T^{3} + 5802 T^{4} + 99 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 8 T + 77 T^{2} - 13 T^{3} + 714 T^{4} - 13 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + T + 74 T^{2} - 68 T^{3} + 3460 T^{4} - 68 p T^{5} + 74 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 12 T + 133 T^{2} - 763 T^{3} + 5768 T^{4} - 763 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 5 T + 126 T^{2} + 568 T^{3} + 7684 T^{4} + 568 p T^{5} + 126 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 5 T + 111 T^{2} - 385 T^{3} + 8011 T^{4} - 385 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 114 T^{2} + 88 T^{3} + 9515 T^{4} + 88 p T^{5} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4 T + 232 T^{2} - 828 T^{3} + 22174 T^{4} - 828 p T^{5} + 232 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 20 T + 375 T^{2} + 4165 T^{3} + 42925 T^{4} + 4165 p T^{5} + 375 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 20 T + 387 T^{2} - 57 p T^{3} + 44118 T^{4} - 57 p^{2} T^{5} + 387 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 17 T + 388 T^{2} + 4009 T^{3} + 48638 T^{4} + 4009 p T^{5} + 388 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 270 T^{2} + 194 T^{3} + 31408 T^{4} + 194 p T^{5} + 270 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 11 T + 266 T^{2} + 1549 T^{3} + 27690 T^{4} + 1549 p T^{5} + 266 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - T + 122 T^{2} - 1096 T^{3} + 12292 T^{4} - 1096 p T^{5} + 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.49619519517840036477797165948, −5.22647917266468075162048015474, −5.12226906379260178962265908387, −4.79646293763180028201366383522, −4.56226506936702915316399510193, −4.42633147690899610238170169119, −4.29229782423517606501374844596, −4.22016515258556046400663224144, −4.01865613232690361214707742783, −3.60962161464734653995688031979, −3.56275660414593274498909694785, −3.43748432181038534560541903929, −3.22755732048517901531965320170, −2.98549848272529657281961996394, −2.68683150592785329526508798278, −2.53586442675143376565979766784, −2.39842534605707709598620779330, −2.31109042727214224982455867084, −1.86559124902586477304134607381, −1.56755634933443402349990924665, −1.41033245981883767143394211320, −1.37072949960239552813855742296, −1.03175658801345090660419141278, −0.43993708233029008854818624015, −0.42699641486266994228717594978,
0.42699641486266994228717594978, 0.43993708233029008854818624015, 1.03175658801345090660419141278, 1.37072949960239552813855742296, 1.41033245981883767143394211320, 1.56755634933443402349990924665, 1.86559124902586477304134607381, 2.31109042727214224982455867084, 2.39842534605707709598620779330, 2.53586442675143376565979766784, 2.68683150592785329526508798278, 2.98549848272529657281961996394, 3.22755732048517901531965320170, 3.43748432181038534560541903929, 3.56275660414593274498909694785, 3.60962161464734653995688031979, 4.01865613232690361214707742783, 4.22016515258556046400663224144, 4.29229782423517606501374844596, 4.42633147690899610238170169119, 4.56226506936702915316399510193, 4.79646293763180028201366383522, 5.12226906379260178962265908387, 5.22647917266468075162048015474, 5.49619519517840036477797165948
