| L(s) = 1 | + 3-s − 3·7-s + 13-s − 3·21-s − 2·25-s − 27-s + 39-s + 43-s + 5·49-s − 61-s + 3·67-s − 2·75-s + 2·79-s − 81-s − 3·91-s − 3·97-s − 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3-s − 3·7-s + 13-s − 3·21-s − 2·25-s − 27-s + 39-s + 43-s + 5·49-s − 61-s + 3·67-s − 2·75-s + 2·79-s − 81-s − 3·91-s − 3·97-s − 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4455603123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4455603123\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
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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
−13.43111917067443547735996430892, −13.11066454330643133217378373806, −12.45387249420584484868636108718, −12.28733419582657825827766868545, −11.41258448051578421724452971916, −10.87660384265640667311897605330, −10.18635749576821935761579769862, −9.667579767592970193870959874302, −9.389480359653358310654892606827, −9.105122860991939043180327958175, −8.223971225401279315001007961967, −7.894287135814863885404921979025, −6.94582858277365901104668311046, −6.59656033850122451083982270329, −5.93563344263929587012050905226, −5.60227233306496467173250173876, −3.99115487950912698917785695222, −3.69870776589072460284248660167, −3.09022774863972417593056188441, −2.35151379692584231952318327679,
2.35151379692584231952318327679, 3.09022774863972417593056188441, 3.69870776589072460284248660167, 3.99115487950912698917785695222, 5.60227233306496467173250173876, 5.93563344263929587012050905226, 6.59656033850122451083982270329, 6.94582858277365901104668311046, 7.894287135814863885404921979025, 8.223971225401279315001007961967, 9.105122860991939043180327958175, 9.389480359653358310654892606827, 9.667579767592970193870959874302, 10.18635749576821935761579769862, 10.87660384265640667311897605330, 11.41258448051578421724452971916, 12.28733419582657825827766868545, 12.45387249420584484868636108718, 13.11066454330643133217378373806, 13.43111917067443547735996430892
