L(s) = 1 | − 2·3-s − 4·4-s − 2·7-s + 9-s + 8·12-s − 8·13-s + 12·16-s + 2·19-s + 4·21-s − 25-s + 4·27-s + 8·28-s − 8·31-s − 4·36-s + 4·37-s + 16·39-s − 2·43-s − 24·48-s − 11·49-s + 32·52-s − 4·57-s − 2·61-s − 2·63-s − 32·64-s − 8·67-s − 14·73-s + 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 2·4-s − 0.755·7-s + 1/3·9-s + 2.30·12-s − 2.21·13-s + 3·16-s + 0.458·19-s + 0.872·21-s − 1/5·25-s + 0.769·27-s + 1.51·28-s − 1.43·31-s − 2/3·36-s + 0.657·37-s + 2.56·39-s − 0.304·43-s − 3.46·48-s − 1.57·49-s + 4.43·52-s − 0.529·57-s − 0.256·61-s − 0.251·63-s − 4·64-s − 0.977·67-s − 1.63·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−12.45987383937424874038080001140, −12.06296001644352266090310246809, −11.37237044309941020825023181952, −10.24320445923116880350231460909, −10.15333178148614777799832547132, −9.238767264318381906771348588364, −9.180168549070062598241568188033, −7.968751144610877316457317025190, −7.43141403312000422638829977686, −6.39084306102520193435942181609, −5.51097905283736056328679977781, −5.03912355415234199771433602492, −4.41466206651142691425988858089, −3.23857552173567549926694025031, 0,
3.23857552173567549926694025031, 4.41466206651142691425988858089, 5.03912355415234199771433602492, 5.51097905283736056328679977781, 6.39084306102520193435942181609, 7.43141403312000422638829977686, 7.968751144610877316457317025190, 9.180168549070062598241568188033, 9.238767264318381906771348588364, 10.15333178148614777799832547132, 10.24320445923116880350231460909, 11.37237044309941020825023181952, 12.06296001644352266090310246809, 12.45987383937424874038080001140