L(s) = 1 | − 9-s + 4·13-s − 2·17-s + 8·19-s + 6·25-s + 16·43-s + 10·49-s + 4·53-s + 8·59-s + 24·67-s + 81-s + 24·83-s − 12·89-s + 12·101-s − 4·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 1.10·13-s − 0.485·17-s + 1.83·19-s + 6/5·25-s + 2.43·43-s + 10/7·49-s + 0.549·53-s + 1.04·59-s + 2.93·67-s + 1/9·81-s + 2.63·83-s − 1.27·89-s + 1.19·101-s − 0.369·117-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.049728610\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.049728610\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−9.351795438020497317530943323698, −9.233227682989218058441581285928, −8.870146758460313143777188392951, −8.409262693197314184207859209367, −8.028852509470346876925921329784, −7.64246566343088870637395789012, −7.05268535986757576902061384124, −6.94182626049642841957424899082, −6.34133884481824026427447758590, −5.89647650793314423395112700273, −5.42722098690632086247899098292, −5.26816652018637832097455929752, −4.55591801305581349750986045922, −4.13090353661117388040202299638, −3.48793154981921838645441341472, −3.34208994975530013596808822052, −2.43942031363122575862104419200, −2.25973780934016763060448245450, −0.956967551606415091539039805849, −0.951200924881774157565969086838,
0.951200924881774157565969086838, 0.956967551606415091539039805849, 2.25973780934016763060448245450, 2.43942031363122575862104419200, 3.34208994975530013596808822052, 3.48793154981921838645441341472, 4.13090353661117388040202299638, 4.55591801305581349750986045922, 5.26816652018637832097455929752, 5.42722098690632086247899098292, 5.89647650793314423395112700273, 6.34133884481824026427447758590, 6.94182626049642841957424899082, 7.05268535986757576902061384124, 7.64246566343088870637395789012, 8.028852509470346876925921329784, 8.409262693197314184207859209367, 8.870146758460313143777188392951, 9.233227682989218058441581285928, 9.351795438020497317530943323698