L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s − 2·7-s − 4·8-s − 2·9-s + 3·12-s + 4·14-s + 5·16-s + 4·18-s + 19-s − 2·21-s − 4·24-s − 10·25-s − 5·27-s − 6·28-s + 18·29-s − 6·32-s − 6·36-s − 2·38-s + 4·42-s + 16·43-s + 5·48-s − 11·49-s + 20·50-s − 6·53-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 2/3·9-s + 0.866·12-s + 1.06·14-s + 5/4·16-s + 0.942·18-s + 0.229·19-s − 0.436·21-s − 0.816·24-s − 2·25-s − 0.962·27-s − 1.13·28-s + 3.34·29-s − 1.06·32-s − 36-s − 0.324·38-s + 0.617·42-s + 2.43·43-s + 0.721·48-s − 1.57·49-s + 2.82·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−8.684413720393242419277779815238, −8.391177552243354804348940653002, −7.73591843742988285704805588989, −7.72023666635394736173695745964, −6.85841529930894935996514770633, −6.52547725848872771498338713632, −5.81027585652647529822118508915, −5.77798205728261117317953598536, −4.61559374588543556536361618627, −4.07392909124551760489549274271, −3.01249692314631579373849129643, −2.99432139105097413830284496966, −2.19179284747142586381309074581, −1.22349109094330361666806860431, 0,
1.22349109094330361666806860431, 2.19179284747142586381309074581, 2.99432139105097413830284496966, 3.01249692314631579373849129643, 4.07392909124551760489549274271, 4.61559374588543556536361618627, 5.77798205728261117317953598536, 5.81027585652647529822118508915, 6.52547725848872771498338713632, 6.85841529930894935996514770633, 7.72023666635394736173695745964, 7.73591843742988285704805588989, 8.391177552243354804348940653002, 8.684413720393242419277779815238