L(s) = 1 | − 2·3-s − 2·4-s + 9-s + 4·12-s + 4·16-s + 4·19-s − 10·25-s + 4·27-s − 2·36-s − 20·43-s − 8·48-s + 14·49-s − 8·57-s − 8·64-s + 28·67-s + 4·73-s + 20·75-s − 8·76-s − 11·81-s − 20·97-s + 20·100-s − 8·108-s + 14·121-s + 127-s + 40·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s + 1/3·9-s + 1.15·12-s + 16-s + 0.917·19-s − 2·25-s + 0.769·27-s − 1/3·36-s − 3.04·43-s − 1.15·48-s + 2·49-s − 1.05·57-s − 64-s + 3.42·67-s + 0.468·73-s + 2.30·75-s − 0.917·76-s − 1.22·81-s − 2.03·97-s + 2·100-s − 0.769·108-s + 1.27·121-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2878147484\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2878147484\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−17.86271258287136632536477627055, −17.71275726124330819883970306751, −16.80125482013019331873847309096, −16.70501923385416248717129689884, −15.64610061145699713347539417384, −15.19730047189340621458262065414, −14.15078414747634405407289801866, −13.75877200386722474683296855826, −13.07485440210465344807068665664, −12.22728533480882669071683978630, −11.75089254705110530455108571385, −11.08980142422841485939985229545, −10.05403923244183463898302503844, −9.720994441844088178423615033944, −8.628873380779201877048120215204, −7.86668938059784971565526833646, −6.71952475783811132454670618982, −5.65243967610745928202997489765, −5.09127124120506387813062870719, −3.79734348170821547926452829026,
3.79734348170821547926452829026, 5.09127124120506387813062870719, 5.65243967610745928202997489765, 6.71952475783811132454670618982, 7.86668938059784971565526833646, 8.628873380779201877048120215204, 9.720994441844088178423615033944, 10.05403923244183463898302503844, 11.08980142422841485939985229545, 11.75089254705110530455108571385, 12.22728533480882669071683978630, 13.07485440210465344807068665664, 13.75877200386722474683296855826, 14.15078414747634405407289801866, 15.19730047189340621458262065414, 15.64610061145699713347539417384, 16.70501923385416248717129689884, 16.80125482013019331873847309096, 17.71275726124330819883970306751, 17.86271258287136632536477627055