L(s) = 1 | + 2-s − 5·7-s + 8-s − 7·11-s − 3·13-s − 5·14-s − 16-s + 17-s − 7·22-s − 3·23-s − 3·26-s + 10·29-s + 2·31-s − 6·32-s + 34-s − 7·37-s + 7·41-s − 15·43-s − 3·46-s − 2·47-s + 8·49-s + 7·53-s − 5·56-s + 10·58-s + 11·59-s + 3·61-s + 2·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.88·7-s + 0.353·8-s − 2.11·11-s − 0.832·13-s − 1.33·14-s − 1/4·16-s + 0.242·17-s − 1.49·22-s − 0.625·23-s − 0.588·26-s + 1.85·29-s + 0.359·31-s − 1.06·32-s + 0.171·34-s − 1.15·37-s + 1.09·41-s − 2.28·43-s − 0.442·46-s − 0.291·47-s + 8/7·49-s + 0.961·53-s − 0.668·56-s + 1.31·58-s + 1.43·59-s + 0.384·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 91 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 145 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 125 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 193 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 169 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.730444298768441764808963038976, −8.628659219548232229305003174605, −8.164038233636233561755920781531, −7.77679408341456702125072508306, −7.23780172080499177658259810480, −6.82153399614620582955752559629, −6.75990196428433942949973990314, −6.14717579549866836141091831724, −5.63273492436235531841763039352, −5.26408427097997828298205429891, −4.97381068423158967279638789608, −4.58050510103992184768453523779, −3.85141864857125002465401942115, −3.66679655155337049898785391655, −2.90679167685540710451014818446, −2.62945711666576056321693141098, −2.37197468164018958408896826304, −1.31271219298675073976653435896, 0, 0,
1.31271219298675073976653435896, 2.37197468164018958408896826304, 2.62945711666576056321693141098, 2.90679167685540710451014818446, 3.66679655155337049898785391655, 3.85141864857125002465401942115, 4.58050510103992184768453523779, 4.97381068423158967279638789608, 5.26408427097997828298205429891, 5.63273492436235531841763039352, 6.14717579549866836141091831724, 6.75990196428433942949973990314, 6.82153399614620582955752559629, 7.23780172080499177658259810480, 7.77679408341456702125072508306, 8.164038233636233561755920781531, 8.628659219548232229305003174605, 8.730444298768441764808963038976