| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 4·11-s + 8·15-s + 12·17-s − 8·19-s + 4·23-s + 4·25-s + 4·27-s + 8·33-s + 8·37-s + 12·41-s + 12·45-s − 8·47-s + 24·51-s − 4·53-s + 16·55-s − 16·57-s + 8·61-s + 8·69-s + 4·71-s + 24·73-s + 8·75-s − 16·79-s + 5·81-s + 8·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s + 2.06·15-s + 2.91·17-s − 1.83·19-s + 0.834·23-s + 4/5·25-s + 0.769·27-s + 1.39·33-s + 1.31·37-s + 1.87·41-s + 1.78·45-s − 1.16·47-s + 3.36·51-s − 0.549·53-s + 2.15·55-s − 2.11·57-s + 1.02·61-s + 0.963·69-s + 0.474·71-s + 2.80·73-s + 0.923·75-s − 1.80·79-s + 5/9·81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.304351607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.304351607\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 4 p T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.259618155289063275474409776343, −9.019247312800431200515920510924, −8.242675091227526195908083913638, −8.233023528620307269063895365542, −7.61104636783699351379699208228, −7.50004668780617771490234564208, −6.64030971592329048853322092672, −6.49293885170379193079393156110, −6.10059447044558534904692846480, −5.80532898145921190008445452790, −5.14670847304765914727283422717, −4.98378436527363291068387131192, −4.18509539851081598699484819786, −3.78830327897411857740174736753, −3.49720142060950376769970905319, −2.81306985594316566180527736166, −2.36855787342397343859960885168, −2.03567067186745051214694189757, −1.16894334555770476561511447763, −1.16469564268184103934275455471,
1.16469564268184103934275455471, 1.16894334555770476561511447763, 2.03567067186745051214694189757, 2.36855787342397343859960885168, 2.81306985594316566180527736166, 3.49720142060950376769970905319, 3.78830327897411857740174736753, 4.18509539851081598699484819786, 4.98378436527363291068387131192, 5.14670847304765914727283422717, 5.80532898145921190008445452790, 6.10059447044558534904692846480, 6.49293885170379193079393156110, 6.64030971592329048853322092672, 7.50004668780617771490234564208, 7.61104636783699351379699208228, 8.233023528620307269063895365542, 8.242675091227526195908083913638, 9.019247312800431200515920510924, 9.259618155289063275474409776343
