L(s) = 1 | − 2·3-s + 9-s + 6·11-s + 6·17-s + 4·19-s − 2·25-s + 4·27-s − 12·33-s − 4·43-s − 2·49-s − 12·51-s − 8·57-s − 12·59-s + 16·67-s + 12·73-s + 4·75-s − 11·81-s + 12·83-s − 12·89-s + 12·97-s + 6·99-s + 18·107-s − 12·113-s + 14·121-s + 127-s + 8·129-s + 131-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.80·11-s + 1.45·17-s + 0.917·19-s − 2/5·25-s + 0.769·27-s − 2.08·33-s − 0.609·43-s − 2/7·49-s − 1.68·51-s − 1.05·57-s − 1.56·59-s + 1.95·67-s + 1.40·73-s + 0.461·75-s − 1.22·81-s + 1.31·83-s − 1.27·89-s + 1.21·97-s + 0.603·99-s + 1.74·107-s − 1.12·113-s + 1.27·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.779085085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779085085\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−7.52747090377821956610605153883, −7.12594626527130568256562505024, −6.61273368560251435639710672123, −6.34133790600429928122816209747, −6.01179034260327607721608996457, −5.49023190868270211652990996329, −5.13846928602031464304778746189, −4.77032540662069658036349049952, −4.14461274373158340937124725800, −3.55649290189302970556073129086, −3.41687728026496463079874815945, −2.63330803552518618426178196434, −1.74655107964577286764543823969, −1.21695768216628226144600501045, −0.64615290850334764621603315778,
0.64615290850334764621603315778, 1.21695768216628226144600501045, 1.74655107964577286764543823969, 2.63330803552518618426178196434, 3.41687728026496463079874815945, 3.55649290189302970556073129086, 4.14461274373158340937124725800, 4.77032540662069658036349049952, 5.13846928602031464304778746189, 5.49023190868270211652990996329, 6.01179034260327607721608996457, 6.34133790600429928122816209747, 6.61273368560251435639710672123, 7.12594626527130568256562505024, 7.52747090377821956610605153883