L(s) = 1 | − 2·3-s − 4-s + 2·5-s + 9-s + 2·12-s − 4·15-s − 3·16-s − 2·20-s + 6·23-s + 3·25-s + 4·27-s − 4·31-s − 36-s − 6·37-s + 2·45-s − 4·47-s + 6·48-s + 6·49-s + 16·53-s + 18·59-s + 4·60-s + 7·64-s − 10·67-s − 12·69-s − 26·71-s − 6·75-s − 6·80-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.894·5-s + 1/3·9-s + 0.577·12-s − 1.03·15-s − 3/4·16-s − 0.447·20-s + 1.25·23-s + 3/5·25-s + 0.769·27-s − 0.718·31-s − 1/6·36-s − 0.986·37-s + 0.298·45-s − 0.583·47-s + 0.866·48-s + 6/7·49-s + 2.19·53-s + 2.34·59-s + 0.516·60-s + 7/8·64-s − 1.22·67-s − 1.44·69-s − 3.08·71-s − 0.692·75-s − 0.670·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.15200678538512294339764483437, −6.98645180238205888047145562793, −6.35238702125101143856676244166, −5.96623564829062141324328272137, −5.63128649801210527937012023724, −5.27495933979172279987143473050, −4.79923915045989238413742381168, −4.61517843593084055733143207930, −3.86166608824258157321503398095, −3.45606466777868687865983718226, −2.64332346339780614042686102246, −2.33073237001605038979795833484, −1.49204935477271765500140522006, −0.876304554922628357154582122613, 0,
0.876304554922628357154582122613, 1.49204935477271765500140522006, 2.33073237001605038979795833484, 2.64332346339780614042686102246, 3.45606466777868687865983718226, 3.86166608824258157321503398095, 4.61517843593084055733143207930, 4.79923915045989238413742381168, 5.27495933979172279987143473050, 5.63128649801210527937012023724, 5.96623564829062141324328272137, 6.35238702125101143856676244166, 6.98645180238205888047145562793, 7.15200678538512294339764483437