L(s) = 1 | − 2·5-s − 2·13-s − 2·17-s + 3·25-s − 2·37-s − 4·41-s − 2·53-s + 4·65-s − 2·73-s + 4·85-s − 2·97-s + 2·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 2·5-s − 2·13-s − 2·17-s + 3·25-s − 2·37-s − 4·41-s − 2·53-s + 4·65-s − 2·73-s + 4·85-s − 2·97-s + 2·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05943376493\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05943376493\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.028902312417474995090016858901, −8.623345293231622288704976911294, −8.483496274461369790474295068571, −7.87050787384386015494758656003, −7.80131188430928146097230568953, −7.04927263170452841074601809363, −7.04464824917153803179918194201, −6.70696618814236765793948525271, −6.41112449763050548042067858941, −5.42282872854658132784515926932, −5.16618443262300700895108142875, −4.71980958619243795448462942401, −4.57197749453620855803271023342, −4.05979962730519690993546485919, −3.59614965549997602557157618609, −2.98002060769753690289434480474, −2.92358627559292952382383428145, −1.98397832221700599696582231182, −1.62252415666222125032276672748, −0.14623722214497856810189978782,
0.14623722214497856810189978782, 1.62252415666222125032276672748, 1.98397832221700599696582231182, 2.92358627559292952382383428145, 2.98002060769753690289434480474, 3.59614965549997602557157618609, 4.05979962730519690993546485919, 4.57197749453620855803271023342, 4.71980958619243795448462942401, 5.16618443262300700895108142875, 5.42282872854658132784515926932, 6.41112449763050548042067858941, 6.70696618814236765793948525271, 7.04464824917153803179918194201, 7.04927263170452841074601809363, 7.80131188430928146097230568953, 7.87050787384386015494758656003, 8.483496274461369790474295068571, 8.623345293231622288704976911294, 9.028902312417474995090016858901