| L(s) = 1 | − 2·3-s − 6·5-s + 14·7-s + 2·9-s − 20·11-s + 18·13-s + 12·15-s + 2·17-s − 28·21-s + 46·23-s + 11·25-s − 18·27-s + 28·31-s + 40·33-s − 84·35-s + 66·37-s − 36·39-s − 28·41-s + 30·43-s − 12·45-s + 78·47-s + 98·49-s − 4·51-s − 14·53-s + 120·55-s + 84·61-s + 28·63-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 6/5·5-s + 2·7-s + 2/9·9-s − 1.81·11-s + 1.38·13-s + 4/5·15-s + 2/17·17-s − 4/3·21-s + 2·23-s + 0.439·25-s − 2/3·27-s + 0.903·31-s + 1.21·33-s − 2.39·35-s + 1.78·37-s − 0.923·39-s − 0.682·41-s + 0.697·43-s − 0.266·45-s + 1.65·47-s + 2·49-s − 0.0784·51-s − 0.264·53-s + 2.18·55-s + 1.37·61-s + 4/9·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.129759227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129759227\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1618 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 30 T + 450 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T + 3042 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3826 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3298 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−14.67535077214901872783188278350, −13.70581060962508907695002622572, −13.33540187767374254163350378370, −12.82899397838247588189098809071, −11.98974020683862037235565252480, −11.61267478054577667802908693450, −11.00550566286039645583661249222, −10.93679536380474073344733669899, −10.39645971781342152745902948694, −9.258201237141827765267687687275, −8.477263128820631142131290060685, −8.039314837497924889158595047933, −7.70398896063801852529497805595, −7.02509083252818418422582478892, −5.89049525383107242513596017353, −5.25112700489574498169748808917, −4.69938114670416991454337161149, −3.96768690193895358555679991101, −2.67840653776458321694065104163, −1.03379330716452026642521066076,
1.03379330716452026642521066076, 2.67840653776458321694065104163, 3.96768690193895358555679991101, 4.69938114670416991454337161149, 5.25112700489574498169748808917, 5.89049525383107242513596017353, 7.02509083252818418422582478892, 7.70398896063801852529497805595, 8.039314837497924889158595047933, 8.477263128820631142131290060685, 9.258201237141827765267687687275, 10.39645971781342152745902948694, 10.93679536380474073344733669899, 11.00550566286039645583661249222, 11.61267478054577667802908693450, 11.98974020683862037235565252480, 12.82899397838247588189098809071, 13.33540187767374254163350378370, 13.70581060962508907695002622572, 14.67535077214901872783188278350
