| L(s) = 1 | − 2.04e3·16-s − 1.08e4·31-s + 2.27e5·61-s − 5.90e4·81-s + 6.44e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 2·16-s − 2.03·31-s + 7.83·61-s − 81-s + 4·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + 1.20e−6·233-s + 1.13e−6·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.102022000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102022000\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T + p^{5} T^{2} )^{2}( 1 + p^{3} T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 523323623 T^{4} + p^{20} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 255564058177 T^{4} + p^{20} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2024677 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2723 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 8665815522315698 T^{4} + p^{20} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1343935055601623 T^{4} + p^{20} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 56927 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 3635284414544796023 T^{4} + p^{20} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 8577821547816235298 T^{4} + p^{20} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 3959005298 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 60206365955391250223 T^{4} + p^{20} T^{8} \) |
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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
−9.658091934250546749838992074836, −9.374592402997372371811841004793, −9.369978548494811638436418643585, −8.663507394245229490727596050850, −8.554464272905495274934849848009, −8.423440454155440039477135249082, −7.997704249738665742659037068857, −7.34676303304615697538010664148, −7.15184695010209829016797227014, −7.05937092040962719869941021280, −6.50866573779947443453840790216, −6.49017387785735636138219013311, −5.61598461060233129412266003543, −5.48003837890953109676474759921, −5.32135353194990800861576382438, −4.63857366517321122128980136421, −4.37064174265297830470859535413, −3.82362106011622147114350609960, −3.69193092548757119495554217209, −3.00904650144693470084630905894, −2.38675278319318177524684093760, −2.10572946469737891106250121282, −1.71588780647384013458755206972, −0.60018037822509827767170569400, −0.55566501642337265539569893758,
0.55566501642337265539569893758, 0.60018037822509827767170569400, 1.71588780647384013458755206972, 2.10572946469737891106250121282, 2.38675278319318177524684093760, 3.00904650144693470084630905894, 3.69193092548757119495554217209, 3.82362106011622147114350609960, 4.37064174265297830470859535413, 4.63857366517321122128980136421, 5.32135353194990800861576382438, 5.48003837890953109676474759921, 5.61598461060233129412266003543, 6.49017387785735636138219013311, 6.50866573779947443453840790216, 7.05937092040962719869941021280, 7.15184695010209829016797227014, 7.34676303304615697538010664148, 7.997704249738665742659037068857, 8.423440454155440039477135249082, 8.554464272905495274934849848009, 8.663507394245229490727596050850, 9.369978548494811638436418643585, 9.374592402997372371811841004793, 9.658091934250546749838992074836
