| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s + 3·13-s + 16-s − 5·17-s + 18-s − 20-s − 4·25-s − 3·26-s − 4·29-s − 32-s + 5·34-s − 36-s − 19·37-s + 40-s + 11·41-s + 45-s + 4·49-s + 4·50-s + 3·52-s + 18·53-s + 4·58-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 1/3·9-s + 0.316·10-s + 0.832·13-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.223·20-s − 4/5·25-s − 0.588·26-s − 0.742·29-s − 0.176·32-s + 0.857·34-s − 1/6·36-s − 3.12·37-s + 0.158·40-s + 1.71·41-s + 0.149·45-s + 4/7·49-s + 0.565·50-s + 0.416·52-s + 2.47·53-s + 0.525·58-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.65458052580328062209533618481, −7.50689570173521597885782495658, −7.04777585105688070547684022815, −6.56788462688372267233975181297, −6.12812898415897558509853125634, −5.63238454742377873934485088839, −5.28938731539157907637239883333, −4.56767011668889777317672089672, −3.98435377068636233473327857951, −3.66219230417062459096527199941, −3.09909697920780680996623621143, −2.20364994341722923598333818130, −1.97915018347050936176831510461, −0.932332448841273221693685064992, 0,
0.932332448841273221693685064992, 1.97915018347050936176831510461, 2.20364994341722923598333818130, 3.09909697920780680996623621143, 3.66219230417062459096527199941, 3.98435377068636233473327857951, 4.56767011668889777317672089672, 5.28938731539157907637239883333, 5.63238454742377873934485088839, 6.12812898415897558509853125634, 6.56788462688372267233975181297, 7.04777585105688070547684022815, 7.50689570173521597885782495658, 7.65458052580328062209533618481
