| L(s) = 1 | − 3·3-s − 5-s − 4·7-s + 3·9-s + 11-s − 4·13-s + 3·15-s − 3·17-s − 5·19-s + 12·21-s − 3·23-s + 5·25-s + 12·29-s − 31-s − 3·33-s + 4·35-s − 5·37-s + 12·39-s − 20·41-s − 8·43-s − 3·45-s + 47-s + 9·49-s + 9·51-s − 9·53-s − 55-s + 15·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 1.51·7-s + 9-s + 0.301·11-s − 1.10·13-s + 0.774·15-s − 0.727·17-s − 1.14·19-s + 2.61·21-s − 0.625·23-s + 25-s + 2.22·29-s − 0.179·31-s − 0.522·33-s + 0.676·35-s − 0.821·37-s + 1.92·39-s − 3.12·41-s − 1.21·43-s − 0.447·45-s + 0.145·47-s + 9/7·49-s + 1.26·51-s − 1.23·53-s − 0.134·55-s + 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−10.61747997355707164025467188021, −10.57740621618608805694527829361, −10.14626422500000299607201316635, −9.815765458038132012689825511468, −9.041946286245670320815761916392, −8.588684008479695694507575685081, −8.340447768786625564727949574523, −7.33799546108765871606028141286, −6.84063791822644202189258750334, −6.73422212067053265184060490740, −6.08473424643705040383204548522, −5.93401584933108306776253388564, −4.90056840148405923987728802608, −4.83097929236300810904347161634, −4.18055981698323905195836329405, −3.21448565480825621746346424175, −2.88487694441176343880457789316, −1.67691603811137524108392664817, 0, 0,
1.67691603811137524108392664817, 2.88487694441176343880457789316, 3.21448565480825621746346424175, 4.18055981698323905195836329405, 4.83097929236300810904347161634, 4.90056840148405923987728802608, 5.93401584933108306776253388564, 6.08473424643705040383204548522, 6.73422212067053265184060490740, 6.84063791822644202189258750334, 7.33799546108765871606028141286, 8.340447768786625564727949574523, 8.588684008479695694507575685081, 9.041946286245670320815761916392, 9.815765458038132012689825511468, 10.14626422500000299607201316635, 10.57740621618608805694527829361, 10.61747997355707164025467188021
