| L(s) = 1 | − 2·3-s − 4-s − 2·5-s + 3·9-s + 2·12-s + 4·15-s − 3·16-s + 2·20-s + 3·25-s − 4·27-s − 16·31-s − 3·36-s − 4·37-s − 6·45-s + 6·48-s − 14·49-s − 12·53-s + 24·59-s − 4·60-s + 7·64-s − 16·67-s + 24·71-s − 6·75-s + 6·80-s + 5·81-s − 12·89-s + 32·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s + 9-s + 0.577·12-s + 1.03·15-s − 3/4·16-s + 0.447·20-s + 3/5·25-s − 0.769·27-s − 2.87·31-s − 1/2·36-s − 0.657·37-s − 0.894·45-s + 0.866·48-s − 2·49-s − 1.64·53-s + 3.12·59-s − 0.516·60-s + 7/8·64-s − 1.95·67-s + 2.84·71-s − 0.692·75-s + 0.670·80-s + 5/9·81-s − 1.27·89-s + 3.31·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.967158660851930466237792642626, −8.820011240823149749756589023974, −8.083568253183071223840268781850, −8.046036996222203290564395408235, −7.38858886797075228904562480866, −6.89974843915113207165478882704, −6.88247408248040237126871603554, −6.33436747418416840785140823830, −5.65189285811065313164211046678, −5.46071292649735554081986056497, −4.82510184431625082527723553300, −4.81308268959775627048620141621, −3.92156562137037026080916460033, −3.86426253102289288200164936510, −3.33025280079733358554184540924, −2.50225461319321110788877891944, −1.82097063686650375861076681109, −1.17998737446998746432223357651, 0, 0,
1.17998737446998746432223357651, 1.82097063686650375861076681109, 2.50225461319321110788877891944, 3.33025280079733358554184540924, 3.86426253102289288200164936510, 3.92156562137037026080916460033, 4.81308268959775627048620141621, 4.82510184431625082527723553300, 5.46071292649735554081986056497, 5.65189285811065313164211046678, 6.33436747418416840785140823830, 6.88247408248040237126871603554, 6.89974843915113207165478882704, 7.38858886797075228904562480866, 8.046036996222203290564395408235, 8.083568253183071223840268781850, 8.820011240823149749756589023974, 8.967158660851930466237792642626
