| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s + 14-s − 16-s − 23-s − 29-s + 35-s − 40-s − 41-s − 2·43-s − 46-s − 47-s + 49-s − 56-s − 58-s + 61-s + 64-s + 67-s + 70-s − 80-s − 82-s − 83-s − 2·86-s + 2·89-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s + 14-s − 16-s − 23-s − 29-s + 35-s − 40-s − 41-s − 2·43-s − 46-s − 47-s + 49-s − 56-s − 58-s + 61-s + 64-s + 67-s + 70-s − 80-s − 82-s − 83-s − 2·86-s + 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.362240344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362240344\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.70225816721088247338415274469, −10.89664684612161506203859970544, −10.23325456326375212320291222532, −10.05384256778273470565226863028, −9.499566059348226330290444137282, −9.146436524116860180669508754909, −8.478473002340239398692614497072, −8.277525887460268212554830754048, −7.70815261552396702511844814822, −7.06821756023737041524021857298, −6.44199065028002407760298267795, −6.18358575565569073566325391242, −5.52900259488754075856356566529, −5.06249212530784345061425221832, −4.97919979098446550649991846714, −4.01063383425975005748374740452, −3.75325010947034271814718782800, −2.93819880043402743187568511838, −2.12142809824458251968121301207, −1.65891548502932067179509079584,
1.65891548502932067179509079584, 2.12142809824458251968121301207, 2.93819880043402743187568511838, 3.75325010947034271814718782800, 4.01063383425975005748374740452, 4.97919979098446550649991846714, 5.06249212530784345061425221832, 5.52900259488754075856356566529, 6.18358575565569073566325391242, 6.44199065028002407760298267795, 7.06821756023737041524021857298, 7.70815261552396702511844814822, 8.277525887460268212554830754048, 8.478473002340239398692614497072, 9.146436524116860180669508754909, 9.499566059348226330290444137282, 10.05384256778273470565226863028, 10.23325456326375212320291222532, 10.89664684612161506203859970544, 11.70225816721088247338415274469
