| L(s) = 1 | − 2-s − 5-s + 2·9-s + 10-s − 11-s − 13-s − 2·18-s − 19-s + 22-s − 23-s + 26-s − 31-s + 32-s + 38-s − 2·45-s + 46-s + 2·49-s + 55-s + 62-s − 64-s + 65-s − 67-s − 73-s + 2·79-s + 3·81-s + 4·83-s − 89-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 2·9-s + 10-s − 11-s − 13-s − 2·18-s − 19-s + 22-s − 23-s + 26-s − 31-s + 32-s + 38-s − 2·45-s + 46-s + 2·49-s + 55-s + 62-s − 64-s + 65-s − 67-s − 73-s + 2·79-s + 3·81-s + 4·83-s − 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1523062674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1523062674\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 79 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−15.09438089440591022273049327848, −14.79301903017583787602292860928, −13.55274723202347412692925946328, −13.49476825961692915439080389739, −12.47946133071320119957371003723, −12.41280998218477785723439677217, −11.81774686271880974772720205144, −10.84468432527915440022069005515, −10.41921929706756148110282082287, −10.00582281033173329485446747233, −9.369771806913826376759339060543, −8.873978905013962342505313744455, −7.87413839665017329527808259336, −7.80057654256192655708865026236, −7.13654942825154090841373972766, −6.38007356073625378647747108423, −5.19349650437579468072287387020, −4.39037048839329186211077735661, −3.84845369197390074372926468934, −2.25042492444233676394569070777,
2.25042492444233676394569070777, 3.84845369197390074372926468934, 4.39037048839329186211077735661, 5.19349650437579468072287387020, 6.38007356073625378647747108423, 7.13654942825154090841373972766, 7.80057654256192655708865026236, 7.87413839665017329527808259336, 8.873978905013962342505313744455, 9.369771806913826376759339060543, 10.00582281033173329485446747233, 10.41921929706756148110282082287, 10.84468432527915440022069005515, 11.81774686271880974772720205144, 12.41280998218477785723439677217, 12.47946133071320119957371003723, 13.49476825961692915439080389739, 13.55274723202347412692925946328, 14.79301903017583787602292860928, 15.09438089440591022273049327848
