L(s) = 1 | − 3·3-s + 4-s + 2·7-s + 6·9-s − 3·12-s − 2·13-s + 16-s + 12·19-s − 6·21-s − 9·25-s − 9·27-s + 2·28-s + 8·31-s + 6·36-s + 6·37-s + 6·39-s − 10·43-s − 3·48-s − 11·49-s − 2·52-s − 36·57-s − 16·61-s + 12·63-s + 64-s − 4·67-s − 20·73-s + 27·75-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1/2·4-s + 0.755·7-s + 2·9-s − 0.866·12-s − 0.554·13-s + 1/4·16-s + 2.75·19-s − 1.30·21-s − 9/5·25-s − 1.73·27-s + 0.377·28-s + 1.43·31-s + 36-s + 0.986·37-s + 0.960·39-s − 1.52·43-s − 0.433·48-s − 1.57·49-s − 0.277·52-s − 4.76·57-s − 2.04·61-s + 1.51·63-s + 1/8·64-s − 0.488·67-s − 2.34·73-s + 3.11·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6467806837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6467806837\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−11.86689967980337109441141028229, −11.49712673239204411601833482255, −11.27108277800122947493852923485, −10.28025762974353066563394220777, −9.945515510046352754787747239151, −9.474403868218177338841530557687, −8.251376244266523722285240612658, −7.45985560667588965978165906144, −7.34105417162701727305699109547, −6.18070949711225184510895261552, −5.88908450915949561478136129218, −4.98797766425562785919630567490, −4.61153453491182141362175201622, −3.20294866149791344283024452913, −1.48472360333277511919704091526,
1.48472360333277511919704091526, 3.20294866149791344283024452913, 4.61153453491182141362175201622, 4.98797766425562785919630567490, 5.88908450915949561478136129218, 6.18070949711225184510895261552, 7.34105417162701727305699109547, 7.45985560667588965978165906144, 8.251376244266523722285240612658, 9.474403868218177338841530557687, 9.945515510046352754787747239151, 10.28025762974353066563394220777, 11.27108277800122947493852923485, 11.49712673239204411601833482255, 11.86689967980337109441141028229