| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 2·13-s + 16-s + 8·19-s + 25-s − 27-s − 16·31-s + 36-s − 20·37-s − 2·39-s + 8·43-s − 48-s − 14·49-s + 2·52-s − 8·57-s − 4·61-s + 64-s − 24·67-s + 4·73-s − 75-s + 8·76-s − 32·79-s + 81-s + 16·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 1.83·19-s + 1/5·25-s − 0.192·27-s − 2.87·31-s + 1/6·36-s − 3.28·37-s − 0.320·39-s + 1.21·43-s − 0.144·48-s − 2·49-s + 0.277·52-s − 1.05·57-s − 0.512·61-s + 1/8·64-s − 2.93·67-s + 0.468·73-s − 0.115·75-s + 0.917·76-s − 3.60·79-s + 1/9·81-s + 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.490403204098803911613209618504, −7.54788789198252757941968322284, −7.45586944398122020573570806182, −7.03582983327921838179279968038, −6.57267675826487089075525795107, −5.88475078175345459986396322172, −5.43160779320744251383504417132, −5.36101206913072414201885147070, −4.53496762516360346181339328954, −3.89256533853822824063870308537, −3.26654328534844814185904778325, −2.98523621593750721047361308357, −1.63917888763865576740316873481, −1.55977141690761434296396620452, 0,
1.55977141690761434296396620452, 1.63917888763865576740316873481, 2.98523621593750721047361308357, 3.26654328534844814185904778325, 3.89256533853822824063870308537, 4.53496762516360346181339328954, 5.36101206913072414201885147070, 5.43160779320744251383504417132, 5.88475078175345459986396322172, 6.57267675826487089075525795107, 7.03582983327921838179279968038, 7.45586944398122020573570806182, 7.54788789198252757941968322284, 8.490403204098803911613209618504
