| L(s) = 1 | − 2·4-s + 2·5-s + 10·7-s − 10·11-s + 4·16-s + 50·17-s + 38·19-s − 4·20-s + 20·23-s − 47·25-s − 20·28-s + 20·35-s + 10·43-s + 20·44-s − 10·47-s − 23·49-s − 20·55-s + 190·61-s − 8·64-s − 100·68-s − 50·73-s − 76·76-s − 100·77-s + 8·80-s + 260·83-s + 100·85-s − 40·92-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/5·5-s + 10/7·7-s − 0.909·11-s + 1/4·16-s + 2.94·17-s + 2·19-s − 1/5·20-s + 0.869·23-s − 1.87·25-s − 5/7·28-s + 4/7·35-s + 0.232·43-s + 5/11·44-s − 0.212·47-s − 0.469·49-s − 0.363·55-s + 3.11·61-s − 1/8·64-s − 1.47·68-s − 0.684·73-s − 76-s − 1.29·77-s + 1/10·80-s + 3.13·83-s + 1.17·85-s − 0.434·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.903177596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903177596\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 122 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1562 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 4970 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3190 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10682 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 358 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18530 T^{2} + p^{4} T^{4} \) |
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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
−11.64064233313356387388896490604, −11.17899327868180286128228551186, −10.49635455337151594836612398517, −10.08031464678942435862383415381, −9.606460989148552492219919178808, −9.547610771394370137802218061984, −8.638209278007247935692835730783, −8.035420270137285824606144895634, −7.75262511500627942824781453916, −7.65496263507166528303072634816, −6.87194462039971213586716194650, −5.82265734008280160214125812000, −5.56134188790578949592862451624, −5.13025967508759334398046374998, −4.84652682556956400263962135709, −3.69898239721956865678086925630, −3.40476709370362647240071995959, −2.48976832081651316792378964185, −1.49924929012738049406820442972, −0.906681318693580108090346085312,
0.906681318693580108090346085312, 1.49924929012738049406820442972, 2.48976832081651316792378964185, 3.40476709370362647240071995959, 3.69898239721956865678086925630, 4.84652682556956400263962135709, 5.13025967508759334398046374998, 5.56134188790578949592862451624, 5.82265734008280160214125812000, 6.87194462039971213586716194650, 7.65496263507166528303072634816, 7.75262511500627942824781453916, 8.035420270137285824606144895634, 8.638209278007247935692835730783, 9.547610771394370137802218061984, 9.606460989148552492219919178808, 10.08031464678942435862383415381, 10.49635455337151594836612398517, 11.17899327868180286128228551186, 11.64064233313356387388896490604
