| L(s) = 1 | + 2-s + 4-s + 8-s − 5·9-s − 2·13-s + 16-s − 5·18-s − 10·25-s − 2·26-s + 32-s − 5·36-s − 8·37-s − 13·49-s − 10·50-s − 2·52-s + 12·53-s + 4·61-s + 64-s − 5·72-s − 2·73-s − 8·74-s + 16·81-s − 6·89-s − 32·97-s − 13·98-s − 10·100-s + 24·101-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s − 0.554·13-s + 1/4·16-s − 1.17·18-s − 2·25-s − 0.392·26-s + 0.176·32-s − 5/6·36-s − 1.31·37-s − 1.85·49-s − 1.41·50-s − 0.277·52-s + 1.64·53-s + 0.512·61-s + 1/8·64-s − 0.589·72-s − 0.234·73-s − 0.929·74-s + 16/9·81-s − 0.635·89-s − 3.24·97-s − 1.31·98-s − 100-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.920152760385419156767633970909, −8.525355978321998752023674487393, −7.79758964924136454445255788566, −7.79510581397524982652968495375, −6.84097212409553947888558604896, −6.52013654011009885219318723813, −5.87360232608420129361802465072, −5.30767778357765891574349430596, −5.26597252709322948443147727457, −4.23471184863465790579846755568, −3.77653001984849480071749037204, −3.05927558579702930204977186926, −2.51874373869269853485475383946, −1.75233952177588605226528654585, 0,
1.75233952177588605226528654585, 2.51874373869269853485475383946, 3.05927558579702930204977186926, 3.77653001984849480071749037204, 4.23471184863465790579846755568, 5.26597252709322948443147727457, 5.30767778357765891574349430596, 5.87360232608420129361802465072, 6.52013654011009885219318723813, 6.84097212409553947888558604896, 7.79510581397524982652968495375, 7.79758964924136454445255788566, 8.525355978321998752023674487393, 8.920152760385419156767633970909
