| L(s) = 1 | + 2·7-s + 6·9-s − 4·17-s + 8·23-s + 10·25-s − 16·31-s + 4·41-s + 3·49-s + 12·63-s + 28·73-s − 8·79-s + 27·81-s + 12·89-s + 12·97-s + 16·103-s + 12·113-s − 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 16·161-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2·9-s − 0.970·17-s + 1.66·23-s + 2·25-s − 2.87·31-s + 0.624·41-s + 3/7·49-s + 1.51·63-s + 3.27·73-s − 0.900·79-s + 3·81-s + 1.27·89-s + 1.21·97-s + 1.57·103-s + 1.12·113-s − 0.733·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 1.26·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.610957465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610957465\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.189786435644848203046447433443, −9.135650725998623285290750233005, −8.878734379490234258949342551132, −8.374125220443382342129593622622, −7.60012811834899552861422360368, −7.59498820807824465502318407863, −7.01832004075914800025445924804, −6.92238324424321597406198806993, −6.44522108096299134878571286584, −5.86926741466675706714715875448, −5.16441386116725114295402616521, −4.92965654473999670807073524902, −4.67946951200927850672233439109, −4.15654526657241875889334656083, −3.55662865214493587313428883551, −3.30223085698716631427980905066, −2.25754894561950131483844248447, −2.06131068526329584299368979408, −1.28389662298130321428503938968, −0.805421380845266839850531709462,
0.805421380845266839850531709462, 1.28389662298130321428503938968, 2.06131068526329584299368979408, 2.25754894561950131483844248447, 3.30223085698716631427980905066, 3.55662865214493587313428883551, 4.15654526657241875889334656083, 4.67946951200927850672233439109, 4.92965654473999670807073524902, 5.16441386116725114295402616521, 5.86926741466675706714715875448, 6.44522108096299134878571286584, 6.92238324424321597406198806993, 7.01832004075914800025445924804, 7.59498820807824465502318407863, 7.60012811834899552861422360368, 8.374125220443382342129593622622, 8.878734379490234258949342551132, 9.135650725998623285290750233005, 9.189786435644848203046447433443
