| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 12-s + 12·13-s − 16-s − 18-s − 8·23-s − 3·24-s − 6·25-s − 12·26-s − 27-s − 5·32-s − 36-s − 20·37-s − 12·39-s + 8·46-s − 24·47-s + 48-s − 14·49-s + 6·50-s − 12·52-s + 54-s + 24·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 3.32·13-s − 1/4·16-s − 0.235·18-s − 1.66·23-s − 0.612·24-s − 6/5·25-s − 2.35·26-s − 0.192·27-s − 0.883·32-s − 1/6·36-s − 3.28·37-s − 1.92·39-s + 1.17·46-s − 3.50·47-s + 0.144·48-s − 2·49-s + 0.848·50-s − 1.66·52-s + 0.136·54-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.700339636508216566718463872719, −8.673289841767403383115719575464, −8.099644677533565214444467559710, −7.962026255736907116683171399764, −6.98236656892345640095319736227, −6.37820814688737049661034336254, −6.30774150521524867405980638808, −5.30285359031178015236684552035, −5.28029214515602867786662657712, −4.16838172205224943400021226417, −3.69904169102384607719527879903, −3.49609663069962101813276746331, −1.72203721066548968493435337820, −1.48488050469889056862198933364, 0,
1.48488050469889056862198933364, 1.72203721066548968493435337820, 3.49609663069962101813276746331, 3.69904169102384607719527879903, 4.16838172205224943400021226417, 5.28029214515602867786662657712, 5.30285359031178015236684552035, 6.30774150521524867405980638808, 6.37820814688737049661034336254, 6.98236656892345640095319736227, 7.962026255736907116683171399764, 8.099644677533565214444467559710, 8.673289841767403383115719575464, 8.700339636508216566718463872719
