| L(s) = 1 | − 2-s − 4-s + 3·8-s + 3·9-s − 6·13-s − 16-s − 6·17-s − 3·18-s − 10·25-s + 6·26-s − 14·29-s − 5·32-s + 6·34-s − 3·36-s + 10·37-s + 12·41-s + 2·49-s + 10·50-s + 6·52-s − 2·53-s + 14·58-s − 16·61-s + 7·64-s + 6·68-s + 9·72-s − 8·73-s − 10·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 9-s − 1.66·13-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 2·25-s + 1.17·26-s − 2.59·29-s − 0.883·32-s + 1.02·34-s − 1/2·36-s + 1.64·37-s + 1.87·41-s + 2/7·49-s + 1.41·50-s + 0.832·52-s − 0.274·53-s + 1.83·58-s − 2.04·61-s + 7/8·64-s + 0.727·68-s + 1.06·72-s − 0.936·73-s − 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−9.816876649626702394475710280684, −9.327704000834677700430666762944, −9.264238826193943461669666199356, −8.391152458618829106017463943911, −7.69699837458779699174173826586, −7.36529032425060881152333666392, −7.20736947999270954805514319970, −6.04347889941024738587947019148, −5.68194003708034400002733283688, −4.52976059179519495486371469698, −4.50862835068296220400083952922, −3.77888204214802338813698524886, −2.43268261083031019191909024135, −1.74674397258489116299867554743, 0,
1.74674397258489116299867554743, 2.43268261083031019191909024135, 3.77888204214802338813698524886, 4.50862835068296220400083952922, 4.52976059179519495486371469698, 5.68194003708034400002733283688, 6.04347889941024738587947019148, 7.20736947999270954805514319970, 7.36529032425060881152333666392, 7.69699837458779699174173826586, 8.391152458618829106017463943911, 9.264238826193943461669666199356, 9.327704000834677700430666762944, 9.816876649626702394475710280684
