| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·8-s − 2·9-s − 6·11-s + 12-s − 16·13-s + 16-s − 17-s − 2·18-s + 5·19-s − 6·22-s + 3·24-s − 16·26-s − 5·27-s + 6·29-s + 4·31-s + 3·32-s − 6·33-s − 34-s − 2·36-s − 9·37-s + 5·38-s − 16·39-s + 13·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.06·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s − 4.43·13-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 1.14·19-s − 1.27·22-s + 0.612·24-s − 3.13·26-s − 0.962·27-s + 1.11·29-s + 0.718·31-s + 0.530·32-s − 1.04·33-s − 0.171·34-s − 1/3·36-s − 1.47·37-s + 0.811·38-s − 2.56·39-s + 2.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.039589057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.039589057\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2^3:S_4$ | \( 1 - T - p T^{3} + p^{2} T^{4} - p^{2} T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - T + p T^{2} - 2 T^{4} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 16 T^{2} + 4 T^{3} + 142 T^{4} + 4 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 6 T + 28 T^{2} + 74 T^{3} + 230 T^{4} + 74 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 16 T + 136 T^{2} + 60 p T^{3} + 3262 T^{4} + 60 p^{2} T^{5} + 136 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + T + 43 T^{2} + 100 T^{3} + 860 T^{4} + 100 p T^{5} + 43 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 5 T + 55 T^{2} - 204 T^{3} + 1476 T^{4} - 204 p T^{5} + 55 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 28 T^{2} - 196 T^{3} + 206 T^{4} - 196 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 6 T + 76 T^{2} - 214 T^{3} + 2270 T^{4} - 214 p T^{5} + 76 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 9 T + 155 T^{2} + 992 T^{3} + 8728 T^{4} + 992 p T^{5} + 155 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 13 T + 181 T^{2} - 1438 T^{3} + 11186 T^{4} - 1438 p T^{5} + 181 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 7 T + 165 T^{2} + 18 p T^{3} + 10278 T^{4} + 18 p^{2} T^{5} + 165 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 14 T + 224 T^{2} - 1918 T^{3} + 16446 T^{4} - 1918 p T^{5} + 224 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 11 T + 137 T^{2} + 646 T^{3} + 6012 T^{4} + 646 p T^{5} + 137 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 13 T + 171 T^{2} - 1676 T^{3} + 15700 T^{4} - 1676 p T^{5} + 171 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 22 T + 388 T^{2} - 4314 T^{3} + 39862 T^{4} - 4314 p T^{5} + 388 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 10 T + 276 T^{2} - 1974 T^{3} + 27974 T^{4} - 1974 p T^{5} + 276 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 3 T + 247 T^{2} - 580 T^{3} + 25376 T^{4} - 580 p T^{5} + 247 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 9 T + 197 T^{2} + 1322 T^{3} + 18556 T^{4} + 1322 p T^{5} + 197 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 16 T + 304 T^{2} + 3292 T^{3} + 35806 T^{4} + 3292 p T^{5} + 304 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 17 T + 329 T^{2} - 3778 T^{3} + 41574 T^{4} - 3778 p T^{5} + 329 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 12 T + 232 T^{2} + 1856 T^{3} + 27110 T^{4} + 1856 p T^{5} + 232 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 16 T + 444 T^{2} + 4608 T^{3} + 67302 T^{4} + 4608 p T^{5} + 444 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.23057097913840350892763533725, −7.19726405796848728403179191180, −7.15494462611920349228618478339, −6.98611549645941963030043093703, −6.46594836957735520020464806887, −6.25821141510975899522210684388, −5.89208456569144413429330069267, −5.71461897221504461218317548952, −5.33460198506086064587598495776, −5.12813290528439509135149945416, −4.97545433602802358636349929294, −4.85376191239317272706649770407, −4.70105145711983626468779465300, −4.44835940749758252217913500955, −3.99700187440753013310003056199, −3.74987406720649996023094647937, −3.21311833957316631922964495343, −2.95870501423024019484989959932, −2.88731332179640666171101225716, −2.59700749192027995524793234873, −2.16573780775582346285068568045, −2.03040231961256133857767186799, −1.98765545700710301856645183912, −0.72809464406958213840134569753, −0.48639576675174269073375387096,
0.48639576675174269073375387096, 0.72809464406958213840134569753, 1.98765545700710301856645183912, 2.03040231961256133857767186799, 2.16573780775582346285068568045, 2.59700749192027995524793234873, 2.88731332179640666171101225716, 2.95870501423024019484989959932, 3.21311833957316631922964495343, 3.74987406720649996023094647937, 3.99700187440753013310003056199, 4.44835940749758252217913500955, 4.70105145711983626468779465300, 4.85376191239317272706649770407, 4.97545433602802358636349929294, 5.12813290528439509135149945416, 5.33460198506086064587598495776, 5.71461897221504461218317548952, 5.89208456569144413429330069267, 6.25821141510975899522210684388, 6.46594836957735520020464806887, 6.98611549645941963030043093703, 7.15494462611920349228618478339, 7.19726405796848728403179191180, 7.23057097913840350892763533725
