L(s) = 1 | − 2·3-s + 4·7-s − 9-s − 2·11-s − 6·17-s − 6·19-s − 8·21-s + 4·23-s + 6·27-s − 16·29-s − 4·31-s + 4·33-s + 4·37-s + 10·41-s − 20·43-s + 8·47-s + 6·49-s + 12·51-s − 4·53-s + 12·57-s − 4·59-s − 12·61-s − 4·63-s + 14·67-s − 8·69-s + 8·71-s + 6·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.51·7-s − 1/3·9-s − 0.603·11-s − 1.45·17-s − 1.37·19-s − 1.74·21-s + 0.834·23-s + 1.15·27-s − 2.97·29-s − 0.718·31-s + 0.696·33-s + 0.657·37-s + 1.56·41-s − 3.04·43-s + 1.16·47-s + 6/7·49-s + 1.68·51-s − 0.549·53-s + 1.58·57-s − 0.520·59-s − 1.53·61-s − 0.503·63-s + 1.71·67-s − 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 147 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 250 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 157 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 251 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.422810110942309047084354172986, −8.196061674467215246783894651855, −7.59536418644360265863608368784, −7.48381147913911507320452885802, −6.95428830316323353464750670744, −6.54738186779889849699436175347, −5.96357974627384299232805309194, −5.95819247217810579139250993485, −5.39311690569094478164847111471, −5.03233215499479208940931208343, −4.63239170883193774814330775956, −4.56268021861056681481632207458, −3.61424755322088549491037936477, −3.60867804581465998381871638343, −2.56269432582653046452508899140, −2.20089270000516141889612229038, −1.86197572659738852506236737575, −1.14635135727810754609995982557, 0, 0,
1.14635135727810754609995982557, 1.86197572659738852506236737575, 2.20089270000516141889612229038, 2.56269432582653046452508899140, 3.60867804581465998381871638343, 3.61424755322088549491037936477, 4.56268021861056681481632207458, 4.63239170883193774814330775956, 5.03233215499479208940931208343, 5.39311690569094478164847111471, 5.95819247217810579139250993485, 5.96357974627384299232805309194, 6.54738186779889849699436175347, 6.95428830316323353464750670744, 7.48381147913911507320452885802, 7.59536418644360265863608368784, 8.196061674467215246783894651855, 8.422810110942309047084354172986