| L(s) = 1 | + 2-s + 4-s + 2·8-s − 9-s + 11-s + 2·16-s − 18-s + 22-s + 23-s − 2·29-s + 2·32-s − 36-s + 37-s − 2·43-s + 44-s + 46-s − 2·53-s − 2·58-s + 3·64-s + 67-s − 2·71-s − 2·72-s + 74-s + 79-s − 2·86-s + 2·88-s + 92-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 2·8-s − 9-s + 11-s + 2·16-s − 18-s + 22-s + 23-s − 2·29-s + 2·32-s − 36-s + 37-s − 2·43-s + 44-s + 46-s − 2·53-s − 2·58-s + 3·64-s + 67-s − 2·71-s − 2·72-s + 74-s + 79-s − 2·86-s + 2·88-s + 92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.251056783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251056783\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.11273830344097394194785571571, −9.818267397437804726328765840831, −9.189403386069815823512362787838, −9.095461305496279478817919965895, −8.212893250609105686544405094365, −8.183465245407622570640834812019, −7.44912845942094310470123251206, −7.33726212827740936246099960690, −6.59593430063734229162133716292, −6.46455745812204634109356258784, −5.93067135229850284971520874692, −5.30824885253189725655032890565, −5.12898918028780398710565239715, −4.58160091886499624170858388547, −4.00807912681082987194580511139, −3.66561247166956056647323094004, −3.09386995085401625795016973948, −2.57891524588464892588207153204, −1.75470996882189676242365278825, −1.36993470125684026314071655134,
1.36993470125684026314071655134, 1.75470996882189676242365278825, 2.57891524588464892588207153204, 3.09386995085401625795016973948, 3.66561247166956056647323094004, 4.00807912681082987194580511139, 4.58160091886499624170858388547, 5.12898918028780398710565239715, 5.30824885253189725655032890565, 5.93067135229850284971520874692, 6.46455745812204634109356258784, 6.59593430063734229162133716292, 7.33726212827740936246099960690, 7.44912845942094310470123251206, 8.183465245407622570640834812019, 8.212893250609105686544405094365, 9.095461305496279478817919965895, 9.189403386069815823512362787838, 9.818267397437804726328765840831, 10.11273830344097394194785571571
