| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s + 4·11-s − 16-s − 17-s + 18-s − 19-s + 4·22-s + 24-s − 2·27-s − 4·33-s − 34-s − 38-s − 2·41-s + 2·43-s + 48-s + 2·49-s + 51-s − 2·54-s + 57-s + 59-s + 64-s − 4·66-s + 2·67-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s + 4·11-s − 16-s − 17-s + 18-s − 19-s + 4·22-s + 24-s − 2·27-s − 4·33-s − 34-s − 38-s − 2·41-s + 2·43-s + 48-s + 2·49-s + 51-s − 2·54-s + 57-s + 59-s + 64-s − 4·66-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.869029596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869029596\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.946511456221940190918718945309, −8.703854960458849041661933009568, −8.302010359068744834789734413390, −7.43967993855636887693117970269, −7.11123658783965316685969147565, −6.91741247388986745227213042868, −6.38966044592526318461200754226, −6.23191229430000748422058641899, −6.06048054694742894635440594044, −5.58511415190801045012047297335, −4.94690648327987914644205900273, −4.67799771275447792930302113568, −4.21335207138998133787515816039, −3.89562819158692057959571383639, −3.75619954034887222552799991625, −3.43691899278919387479246264963, −2.20961884829812915988691692816, −2.14832972380656887458301148298, −1.31321151694277335264848694313, −0.799358572018411046409269381166,
0.799358572018411046409269381166, 1.31321151694277335264848694313, 2.14832972380656887458301148298, 2.20961884829812915988691692816, 3.43691899278919387479246264963, 3.75619954034887222552799991625, 3.89562819158692057959571383639, 4.21335207138998133787515816039, 4.67799771275447792930302113568, 4.94690648327987914644205900273, 5.58511415190801045012047297335, 6.06048054694742894635440594044, 6.23191229430000748422058641899, 6.38966044592526318461200754226, 6.91741247388986745227213042868, 7.11123658783965316685969147565, 7.43967993855636887693117970269, 8.302010359068744834789734413390, 8.703854960458849041661933009568, 8.946511456221940190918718945309
