| L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s − 8-s + 6·9-s − 4·12-s + 16-s + 12·17-s − 6·18-s + 2·19-s + 4·24-s − 10·25-s + 4·27-s − 8·31-s − 32-s − 12·34-s + 6·36-s − 2·38-s − 4·48-s + 49-s + 10·50-s − 48·51-s − 4·54-s − 8·57-s − 12·59-s + 16·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.15·12-s + 1/4·16-s + 2.91·17-s − 1.41·18-s + 0.458·19-s + 0.816·24-s − 2·25-s + 0.769·27-s − 1.43·31-s − 0.176·32-s − 2.05·34-s + 36-s − 0.324·38-s − 0.577·48-s + 1/7·49-s + 1.41·50-s − 6.72·51-s − 0.544·54-s − 1.05·57-s − 1.56·59-s + 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 566048 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 566048 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.081070482945495560550014170392, −7.57571100088867902110310233811, −7.47409117202170347395035873500, −6.75439540759619740939389014884, −6.21485262589367409754991816651, −5.86507697049146782198264869749, −5.57928681742950427486583645839, −5.23719852670262600297208829240, −4.77780388213807797595021740659, −3.72379689627192733903633245428, −3.49672951871907610112656515855, −2.56658671720506586948973480294, −1.53114392297898773356855355003, −0.905061494360852486185250338719, 0,
0.905061494360852486185250338719, 1.53114392297898773356855355003, 2.56658671720506586948973480294, 3.49672951871907610112656515855, 3.72379689627192733903633245428, 4.77780388213807797595021740659, 5.23719852670262600297208829240, 5.57928681742950427486583645839, 5.86507697049146782198264869749, 6.21485262589367409754991816651, 6.75439540759619740939389014884, 7.47409117202170347395035873500, 7.57571100088867902110310233811, 8.081070482945495560550014170392
