| L(s) = 1 | − 128·16-s − 1.23e3·31-s + 728·61-s − 729·81-s + 5.32e3·121-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 − 7.13·31-s + 1.52·61-s − 81-s + 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3735036274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3735036274\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )^{2}( 1 + p^{2} T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 153502 T^{4} + p^{12} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 9397582 T^{4} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10582 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 2826257618 T^{4} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 235885102 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 151031344462 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 104459767778 T^{4} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 204622 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 1662757858942 T^{4} + p^{12} T^{8} \) |
| 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
−10.12385170028296534938851697431, −10.11950423313335467063745431566, −9.536167660706677099755196055713, −9.258744457525703046406363256917, −9.074637579112543327632432122443, −8.958559805647251843468297307401, −8.428831700997192317029759745771, −8.342099669103551740273421157770, −7.44224705265369838699696619442, −7.41399193393128351082120012605, −7.23429046704820288803202808301, −6.99512621823117408623741034430, −6.45187164590044599683944045112, −6.02363246610559724058335219709, −5.65164556792110258427606561309, −5.27713504371726710045516538650, −5.10951789879178630656387643032, −4.47692402468959738687048463562, −4.03940896076538959345397896946, −3.57917861674967906275874497601, −3.43298453718427432762794515768, −2.42209572897341650854255784092, −2.04844984145428904494340941490, −1.61366602085828627509237671951, −0.20214257444513072306574759506,
0.20214257444513072306574759506, 1.61366602085828627509237671951, 2.04844984145428904494340941490, 2.42209572897341650854255784092, 3.43298453718427432762794515768, 3.57917861674967906275874497601, 4.03940896076538959345397896946, 4.47692402468959738687048463562, 5.10951789879178630656387643032, 5.27713504371726710045516538650, 5.65164556792110258427606561309, 6.02363246610559724058335219709, 6.45187164590044599683944045112, 6.99512621823117408623741034430, 7.23429046704820288803202808301, 7.41399193393128351082120012605, 7.44224705265369838699696619442, 8.342099669103551740273421157770, 8.428831700997192317029759745771, 8.958559805647251843468297307401, 9.074637579112543327632432122443, 9.258744457525703046406363256917, 9.536167660706677099755196055713, 10.11950423313335467063745431566, 10.12385170028296534938851697431
