| L(s) = 1 | + 2-s + 4·3-s + 4-s + 4·6-s + 8-s + 6·9-s + 4·12-s + 16-s + 12·17-s + 6·18-s − 2·19-s + 4·24-s − 10·25-s − 4·27-s + 8·31-s + 32-s + 12·34-s + 6·36-s − 2·38-s + 4·48-s + 49-s − 10·50-s + 48·51-s − 4·54-s − 8·57-s + 12·59-s + 16·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.353·8-s + 2·9-s + 1.15·12-s + 1/4·16-s + 2.91·17-s + 1.41·18-s − 0.458·19-s + 0.816·24-s − 2·25-s − 0.769·27-s + 1.43·31-s + 0.176·32-s + 2.05·34-s + 36-s − 0.324·38-s + 0.577·48-s + 1/7·49-s − 1.41·50-s + 6.72·51-s − 0.544·54-s − 1.05·57-s + 1.56·59-s + 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 566048 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 566048 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.255200656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.255200656\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.196197307405668368216926690290, −8.182643116839158569055135937831, −7.72200516209190044787449977815, −7.29668061464756137953129595214, −6.77298005536370046551425088692, −5.97907088754740589421145166260, −5.62168100149868099368912779374, −5.31155422432102619146983664079, −4.35906178911561015019581069777, −3.82418745638366191622617161116, −3.55246093072316681260396868353, −3.09738642858536778661614069445, −2.48514181668809031419660323366, −2.11458272020859388026266946767, −1.19460132813703026742735308794,
1.19460132813703026742735308794, 2.11458272020859388026266946767, 2.48514181668809031419660323366, 3.09738642858536778661614069445, 3.55246093072316681260396868353, 3.82418745638366191622617161116, 4.35906178911561015019581069777, 5.31155422432102619146983664079, 5.62168100149868099368912779374, 5.97907088754740589421145166260, 6.77298005536370046551425088692, 7.29668061464756137953129595214, 7.72200516209190044787449977815, 8.182643116839158569055135937831, 8.196197307405668368216926690290
