| L(s) = 1 | − 2·4-s − 2·9-s + 12·11-s + 4·16-s − 12·17-s − 2·19-s + 10·25-s + 4·36-s + 20·43-s − 24·44-s + 14·49-s − 8·64-s + 24·68-s − 4·73-s + 4·76-s − 5·81-s − 36·83-s − 24·99-s − 20·100-s + 86·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 3.61·11-s + 16-s − 2.91·17-s − 0.458·19-s + 2·25-s + 2/3·36-s + 3.04·43-s − 3.61·44-s + 2·49-s − 64-s + 2.91·68-s − 0.468·73-s + 0.458·76-s − 5/9·81-s − 3.95·83-s − 2.41·99-s − 2·100-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.029293857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.029293857\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−14.06603944287786377715490547402, −12.52435652665062125255390928256, −12.49764582382847825180827518004, −11.67323605857863758382696666181, −11.20165668185067647345703101711, −10.97613773593511041358776305788, −10.15683724626665780851439835791, −9.231060255289630134501560476554, −9.114205830842331838226863030820, −8.713463771272228014018315413965, −8.595889737809797445841617266431, −7.13176191494517566213060331647, −6.92076049135155127354397533378, −6.22621922361415972495014933359, −5.80077049785098798397585628347, −4.46200309278818235954587794442, −4.36676230116290252808771263576, −3.80110313522409954107472059204, −2.56898448051577982608925901746, −1.17299461409151067318190036894,
1.17299461409151067318190036894, 2.56898448051577982608925901746, 3.80110313522409954107472059204, 4.36676230116290252808771263576, 4.46200309278818235954587794442, 5.80077049785098798397585628347, 6.22621922361415972495014933359, 6.92076049135155127354397533378, 7.13176191494517566213060331647, 8.595889737809797445841617266431, 8.713463771272228014018315413965, 9.114205830842331838226863030820, 9.231060255289630134501560476554, 10.15683724626665780851439835791, 10.97613773593511041358776305788, 11.20165668185067647345703101711, 11.67323605857863758382696666181, 12.49764582382847825180827518004, 12.52435652665062125255390928256, 14.06603944287786377715490547402
