| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s + 11-s − 15-s − 16-s − 22-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 31-s + 33-s − 40-s − 48-s − 50-s + 53-s + 54-s − 55-s + 2·58-s − 59-s + 62-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s + 11-s − 15-s − 16-s − 22-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 31-s + 33-s − 40-s − 48-s − 50-s + 53-s + 54-s − 55-s + 2·58-s − 59-s + 62-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6025527996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6025527996\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.11649467267522829734197393932, −9.460920323510715531673400701629, −9.185071813372851386745157497706, −8.907037776560684756546142192240, −8.820130259655560051029766663259, −8.123674128320539566041894563446, −7.75414314137893028240538963046, −7.48543079372627703802208122715, −7.32380998864990173117437317453, −6.42905678795096232028728501384, −6.34137612611786791548647542959, −5.30194883798291579962600069860, −5.16674276485991631866735976195, −4.33627918375669845845352337523, −3.95038864210435977563621553114, −3.53830671774421151814635561205, −3.22029161399103696854180876123, −2.10145150397294409097414992252, −1.92269843499679796701520053419, −0.796102723187690449342926660761,
0.796102723187690449342926660761, 1.92269843499679796701520053419, 2.10145150397294409097414992252, 3.22029161399103696854180876123, 3.53830671774421151814635561205, 3.95038864210435977563621553114, 4.33627918375669845845352337523, 5.16674276485991631866735976195, 5.30194883798291579962600069860, 6.34137612611786791548647542959, 6.42905678795096232028728501384, 7.32380998864990173117437317453, 7.48543079372627703802208122715, 7.75414314137893028240538963046, 8.123674128320539566041894563446, 8.820130259655560051029766663259, 8.907037776560684756546142192240, 9.185071813372851386745157497706, 9.460920323510715531673400701629, 10.11649467267522829734197393932
