| L(s) = 1 | − 3·3-s − 4-s − 7-s + 6·9-s + 3·12-s + 3·13-s + 16-s + 3·21-s − 25-s − 9·27-s + 28-s + 9·31-s − 6·36-s + 5·37-s − 9·39-s + 8·43-s − 3·48-s + 49-s − 3·52-s + 9·61-s − 6·63-s − 64-s − 5·67-s + 73-s + 3·75-s + 4·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s − 0.377·7-s + 2·9-s + 0.866·12-s + 0.832·13-s + 1/4·16-s + 0.654·21-s − 1/5·25-s − 1.73·27-s + 0.188·28-s + 1.61·31-s − 36-s + 0.821·37-s − 1.44·39-s + 1.21·43-s − 0.433·48-s + 1/7·49-s − 0.416·52-s + 1.15·61-s − 0.755·63-s − 1/8·64-s − 0.610·67-s + 0.117·73-s + 0.346·75-s + 0.450·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9117869335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9117869335\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.318008438991289993315007888666, −7.48245353184456152941549350213, −7.35156564777050085522577847765, −6.60638182693326161618251056166, −6.31635538667819651049251483209, −5.94468201904435523015631772120, −5.61239355675081315079617672570, −5.01077757322795603322785085211, −4.56134135438064287025250025384, −4.19938866623469113746777941571, −3.63784553140918674542066302688, −2.95429116618460432717681410796, −2.11105866831813092685590283972, −1.12106303563957554716196069464, −0.61577346227566000338851588569,
0.61577346227566000338851588569, 1.12106303563957554716196069464, 2.11105866831813092685590283972, 2.95429116618460432717681410796, 3.63784553140918674542066302688, 4.19938866623469113746777941571, 4.56134135438064287025250025384, 5.01077757322795603322785085211, 5.61239355675081315079617672570, 5.94468201904435523015631772120, 6.31635538667819651049251483209, 6.60638182693326161618251056166, 7.35156564777050085522577847765, 7.48245353184456152941549350213, 8.318008438991289993315007888666
