| L(s) = 1 | + 3-s − 4-s − 3·7-s + 9-s − 12-s + 3·13-s − 3·16-s − 8·19-s − 3·21-s + 2·25-s + 27-s + 3·28-s + 4·31-s − 36-s − 8·37-s + 3·39-s + 16·43-s − 3·48-s + 6·49-s − 3·52-s − 8·57-s + 16·61-s − 3·63-s + 7·64-s − 8·67-s + 4·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 1.13·7-s + 1/3·9-s − 0.288·12-s + 0.832·13-s − 3/4·16-s − 1.83·19-s − 0.654·21-s + 2/5·25-s + 0.192·27-s + 0.566·28-s + 0.718·31-s − 1/6·36-s − 1.31·37-s + 0.480·39-s + 2.43·43-s − 0.433·48-s + 6/7·49-s − 0.416·52-s − 1.05·57-s + 2.04·61-s − 0.377·63-s + 7/8·64-s − 0.977·67-s + 0.468·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6828945434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6828945434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−13.15462021230623033187473107990, −12.64296706297696533573800696145, −12.09416452733982837357248399870, −11.03875658285883297561866499921, −10.61127167204518793650702346210, −9.881797812406761228139013042516, −9.196397174384002639829606535855, −8.719229117878705718700334994389, −8.215896790738348096823310012783, −7.05004095104865533374700080458, −6.55535484914200555567754755910, −5.71774746168000328000796578581, −4.43184143650793104046710884591, −3.78357957995193524574068323677, −2.54663383021793368454863755150,
2.54663383021793368454863755150, 3.78357957995193524574068323677, 4.43184143650793104046710884591, 5.71774746168000328000796578581, 6.55535484914200555567754755910, 7.05004095104865533374700080458, 8.215896790738348096823310012783, 8.719229117878705718700334994389, 9.196397174384002639829606535855, 9.881797812406761228139013042516, 10.61127167204518793650702346210, 11.03875658285883297561866499921, 12.09416452733982837357248399870, 12.64296706297696533573800696145, 13.15462021230623033187473107990
