L(s) = 1 | − 2-s − 4-s − 6·5-s + 3·8-s − 2·9-s + 6·10-s + 2·13-s − 16-s + 8·17-s + 2·18-s + 6·20-s + 17·25-s − 2·26-s − 12·29-s − 5·32-s − 8·34-s + 2·36-s + 16·37-s − 18·40-s + 10·41-s + 12·45-s − 13·49-s − 17·50-s − 2·52-s + 12·53-s + 12·58-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 2.68·5-s + 1.06·8-s − 2/3·9-s + 1.89·10-s + 0.554·13-s − 1/4·16-s + 1.94·17-s + 0.471·18-s + 1.34·20-s + 17/5·25-s − 0.392·26-s − 2.22·29-s − 0.883·32-s − 1.37·34-s + 1/3·36-s + 2.63·37-s − 2.84·40-s + 1.56·41-s + 1.78·45-s − 1.85·49-s − 2.40·50-s − 0.277·52-s + 1.64·53-s + 1.57·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.485054135160731089102284191623, −9.284736779356232209822876415615, −8.305026714024089359524101699318, −8.287570693733701613432166498970, −7.66039763333771123227596439320, −7.60066555070115619075552893577, −6.96431333752530540741243212893, −5.71986854568663317974018114506, −5.60157612088078313975836309422, −4.29537092118666054657945621631, −4.25666915587159511841100490901, −3.55965142579005276204406242229, −2.95169562046434992926302882335, −1.13914830433108299371056120620, 0,
1.13914830433108299371056120620, 2.95169562046434992926302882335, 3.55965142579005276204406242229, 4.25666915587159511841100490901, 4.29537092118666054657945621631, 5.60157612088078313975836309422, 5.71986854568663317974018114506, 6.96431333752530540741243212893, 7.60066555070115619075552893577, 7.66039763333771123227596439320, 8.287570693733701613432166498970, 8.305026714024089359524101699318, 9.284736779356232209822876415615, 9.485054135160731089102284191623