| L(s) = 1 | − 2-s − 4-s − 5·5-s + 3·8-s − 9-s + 5·10-s − 10·13-s − 16-s + 17-s + 18-s + 5·20-s + 11·25-s + 10·26-s + 6·29-s − 5·32-s − 34-s + 36-s − 7·37-s − 15·40-s + 12·41-s + 5·45-s − 49-s − 11·50-s + 10·52-s − 14·53-s − 6·58-s + 3·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 2.23·5-s + 1.06·8-s − 1/3·9-s + 1.58·10-s − 2.77·13-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.11·20-s + 11/5·25-s + 1.96·26-s + 1.11·29-s − 0.883·32-s − 0.171·34-s + 1/6·36-s − 1.15·37-s − 2.37·40-s + 1.87·41-s + 0.745·45-s − 1/7·49-s − 1.55·50-s + 1.38·52-s − 1.92·53-s − 0.787·58-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 277 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 22 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 135 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
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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
−12.18221992927408999265138115878, −11.56420403389407839470950712632, −10.91109751717384157204938798506, −10.26902095131434557327771296406, −9.626896639661071213681566045074, −9.088435107566179982389963273317, −8.201102724093229098697436247525, −7.85795817417507767421079023204, −7.43866433215602758386987900462, −6.81988355466730702910757400569, −5.26290023619126605185058658121, −4.62787660593522981488347226462, −4.03134283450707958081814326856, −2.84974722984050131712504061599, 0,
2.84974722984050131712504061599, 4.03134283450707958081814326856, 4.62787660593522981488347226462, 5.26290023619126605185058658121, 6.81988355466730702910757400569, 7.43866433215602758386987900462, 7.85795817417507767421079023204, 8.201102724093229098697436247525, 9.088435107566179982389963273317, 9.626896639661071213681566045074, 10.26902095131434557327771296406, 10.91109751717384157204938798506, 11.56420403389407839470950712632, 12.18221992927408999265138115878
