| L(s) = 1 | − 2-s − 5-s + 7-s + 8-s + 10-s − 2·11-s + 13-s − 14-s − 16-s − 2·19-s + 2·22-s + 23-s − 26-s − 35-s + 4·37-s + 2·38-s − 40-s + 41-s − 46-s + 47-s + 49-s − 2·53-s + 2·55-s + 56-s + 59-s + 64-s − 65-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 7-s + 8-s + 10-s − 2·11-s + 13-s − 14-s − 16-s − 2·19-s + 2·22-s + 23-s − 26-s − 35-s + 4·37-s + 2·38-s − 40-s + 41-s − 46-s + 47-s + 49-s − 2·53-s + 2·55-s + 56-s + 59-s + 64-s − 65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5640359355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5640359355\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−8.942633923580969335391689214883, −8.480676807611327247294799801035, −8.209373712572557973251502358896, −8.003579776703579370758019413072, −7.63357820498692989287934724390, −7.58152837765408716159878823074, −7.03238605825532987182737946157, −6.34197433296330745668163290648, −6.10226298989492232937594995415, −5.68749489108613563593076449986, −5.03957871765039858921651706046, −4.65063956843494897380920665463, −4.52720176469718369979864929211, −4.02992299739737853962169585471, −3.62347378956749057768147089426, −2.85468991959902631254159574940, −2.35505118207077451704885693061, −2.10995009594499634705880901142, −1.14257753522792945873190143295, −0.63088498262957786877999365299,
0.63088498262957786877999365299, 1.14257753522792945873190143295, 2.10995009594499634705880901142, 2.35505118207077451704885693061, 2.85468991959902631254159574940, 3.62347378956749057768147089426, 4.02992299739737853962169585471, 4.52720176469718369979864929211, 4.65063956843494897380920665463, 5.03957871765039858921651706046, 5.68749489108613563593076449986, 6.10226298989492232937594995415, 6.34197433296330745668163290648, 7.03238605825532987182737946157, 7.58152837765408716159878823074, 7.63357820498692989287934724390, 8.003579776703579370758019413072, 8.209373712572557973251502358896, 8.480676807611327247294799801035, 8.942633923580969335391689214883
