L(s) = 1 | − 2·3-s + 4·7-s + 2·9-s − 12·13-s − 8·21-s + 10·25-s − 6·27-s − 16·31-s − 12·37-s + 24·39-s + 12·43-s + 8·49-s − 24·61-s + 8·63-s + 4·67-s + 4·73-s − 20·75-s + 11·81-s − 48·91-s + 32·93-s + 36·97-s − 4·103-s + 24·111-s − 24·117-s + 4·121-s + 127-s − 24·129-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.51·7-s + 2/3·9-s − 3.32·13-s − 1.74·21-s + 2·25-s − 1.15·27-s − 2.87·31-s − 1.97·37-s + 3.84·39-s + 1.82·43-s + 8/7·49-s − 3.07·61-s + 1.00·63-s + 0.488·67-s + 0.468·73-s − 2.30·75-s + 11/9·81-s − 5.03·91-s + 3.31·93-s + 3.65·97-s − 0.394·103-s + 2.27·111-s − 2.21·117-s + 4/11·121-s + 0.0887·127-s − 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6768946904\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6768946904\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 238 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2^3$ | \( 1 + 3598 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.919607865391245298275897633150, −8.806836119434690603895943353598, −8.256690120174967706303580531604, −7.81342454328930457337331884930, −7.80532451726425038498186781396, −7.38320574069158203054610038036, −7.31919772017105143241429393204, −7.13510109905449442202161937971, −6.85262402447856767543863738183, −6.50822227822480342619269858947, −5.96885691089216655569722561515, −5.77208246777687827150158630480, −5.51593829812830579038286024180, −5.16813958803800113904499677103, −4.97374432583063820940650950247, −4.77540700702460406257143055254, −4.54897954160910917238448004036, −4.25062191776734368967577281914, −3.60271592086113855295189830632, −3.32113056233185999071783892220, −2.81366727372100398059723332847, −2.23804720887372384460904940647, −1.99209886100742048793905933810, −1.57657519551515762396399213495, −0.47419584915157173845336446608,
0.47419584915157173845336446608, 1.57657519551515762396399213495, 1.99209886100742048793905933810, 2.23804720887372384460904940647, 2.81366727372100398059723332847, 3.32113056233185999071783892220, 3.60271592086113855295189830632, 4.25062191776734368967577281914, 4.54897954160910917238448004036, 4.77540700702460406257143055254, 4.97374432583063820940650950247, 5.16813958803800113904499677103, 5.51593829812830579038286024180, 5.77208246777687827150158630480, 5.96885691089216655569722561515, 6.50822227822480342619269858947, 6.85262402447856767543863738183, 7.13510109905449442202161937971, 7.31919772017105143241429393204, 7.38320574069158203054610038036, 7.80532451726425038498186781396, 7.81342454328930457337331884930, 8.256690120174967706303580531604, 8.806836119434690603895943353598, 8.919607865391245298275897633150