| L(s) = 1 | + 4-s + 2·7-s + 2·28-s − 2·37-s − 2·43-s + 3·49-s − 64-s + 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s + 2·7-s + 2·28-s − 2·37-s − 2·43-s + 3·49-s − 64-s + 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.839449936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839449936\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−10.02944591943552765476781696696, −9.349615901642857061912383115461, −8.902902054120991952451699376794, −8.424769593532622569408273807934, −8.301686865379305276477974453446, −7.932734379283643434734382269335, −7.24880537699402714312724214322, −6.99940598940526418624129680332, −6.93558457106526796701733672629, −6.05763279934618195094641238203, −5.80869347817437297985492816577, −5.23210941933012756997585437601, −4.79922409943941392386911840550, −4.63449302403676709744177095760, −3.84928487291438368783595224287, −3.41769352103128106380981335470, −2.76184147861209524498395113499, −1.96985727032657027268391864894, −1.93960480328905945514688461578, −1.20585865558191633564296078501,
1.20585865558191633564296078501, 1.93960480328905945514688461578, 1.96985727032657027268391864894, 2.76184147861209524498395113499, 3.41769352103128106380981335470, 3.84928487291438368783595224287, 4.63449302403676709744177095760, 4.79922409943941392386911840550, 5.23210941933012756997585437601, 5.80869347817437297985492816577, 6.05763279934618195094641238203, 6.93558457106526796701733672629, 6.99940598940526418624129680332, 7.24880537699402714312724214322, 7.932734379283643434734382269335, 8.301686865379305276477974453446, 8.424769593532622569408273807934, 8.902902054120991952451699376794, 9.349615901642857061912383115461, 10.02944591943552765476781696696
