L(s) = 1 | − 2·3-s + 4·5-s + 9-s − 8·15-s + 12·17-s + 6·25-s + 4·27-s + 4·37-s + 4·41-s + 4·43-s + 4·45-s − 8·47-s − 7·49-s − 24·51-s + 4·59-s + 12·67-s − 12·75-s + 24·79-s − 11·81-s − 20·83-s + 48·85-s − 4·89-s − 12·101-s − 12·109-s − 8·111-s + 10·121-s − 8·123-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s − 2.06·15-s + 2.91·17-s + 6/5·25-s + 0.769·27-s + 0.657·37-s + 0.624·41-s + 0.609·43-s + 0.596·45-s − 1.16·47-s − 49-s − 3.36·51-s + 0.520·59-s + 1.46·67-s − 1.38·75-s + 2.70·79-s − 1.22·81-s − 2.19·83-s + 5.20·85-s − 0.423·89-s − 1.19·101-s − 1.14·109-s − 0.759·111-s + 0.909·121-s − 0.721·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.989534943\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.989534943\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−8.500613867401517393357255463291, −8.061497246035729874104572559646, −7.74122434240927649838749671472, −6.95931609134650597347863493297, −6.63338750661396004135383717830, −6.03855628963688194539895681153, −5.70099290630377806442347090102, −5.51675301087653101768672298689, −5.07478642145238487372542719776, −4.45152867476489909725205635557, −3.57412951982818749504576517822, −3.03861996821360169451677488107, −2.32575459300543708723025227175, −1.49607957713378411683379389758, −0.899195721352296982487281372586,
0.899195721352296982487281372586, 1.49607957713378411683379389758, 2.32575459300543708723025227175, 3.03861996821360169451677488107, 3.57412951982818749504576517822, 4.45152867476489909725205635557, 5.07478642145238487372542719776, 5.51675301087653101768672298689, 5.70099290630377806442347090102, 6.03855628963688194539895681153, 6.63338750661396004135383717830, 6.95931609134650597347863493297, 7.74122434240927649838749671472, 8.061497246035729874104572559646, 8.500613867401517393357255463291