| L(s) = 1 | − 2-s − 3·3-s − 4-s + 3·6-s + 3·8-s + 4·9-s − 8·11-s + 3·12-s − 16-s + 2·17-s − 4·18-s + 2·19-s + 8·22-s − 9·24-s + 8·25-s − 5·32-s + 24·33-s − 2·34-s − 4·36-s − 2·38-s − 8·41-s + 2·43-s + 8·44-s + 3·48-s + 4·49-s − 8·50-s − 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.22·6-s + 1.06·8-s + 4/3·9-s − 2.41·11-s + 0.866·12-s − 1/4·16-s + 0.485·17-s − 0.942·18-s + 0.458·19-s + 1.70·22-s − 1.83·24-s + 8/5·25-s − 0.883·32-s + 4.17·33-s − 0.342·34-s − 2/3·36-s − 0.324·38-s − 1.24·41-s + 0.304·43-s + 1.20·44-s + 0.433·48-s + 4/7·49-s − 1.13·50-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 353 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 18 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.879632125109081838290292690218, −9.176067736057798807197415054598, −8.481323577524403977832707480137, −8.181280079132349162767504132275, −7.54951765382461205957888806043, −7.11182934380248060561947404641, −6.51090383934320497007336696758, −5.67969573672892041377748666207, −5.27645315142293311187475674974, −5.05098516327876625832392129371, −4.46118950077932848094713516441, −3.36150910872468018995886601382, −2.48635744538874003342307300442, −1.04917504180427841827509159832, 0,
1.04917504180427841827509159832, 2.48635744538874003342307300442, 3.36150910872468018995886601382, 4.46118950077932848094713516441, 5.05098516327876625832392129371, 5.27645315142293311187475674974, 5.67969573672892041377748666207, 6.51090383934320497007336696758, 7.11182934380248060561947404641, 7.54951765382461205957888806043, 8.181280079132349162767504132275, 8.481323577524403977832707480137, 9.176067736057798807197415054598, 9.879632125109081838290292690218
