L(s) = 1 | − 8·7-s − 3·9-s + 8·19-s + 10·25-s − 16·43-s + 34·49-s − 28·61-s + 24·63-s − 20·73-s + 9·81-s + 22·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·171-s + 173-s − 80·175-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 3.02·7-s − 9-s + 1.83·19-s + 2·25-s − 2.43·43-s + 34/7·49-s − 3.58·61-s + 3.02·63-s − 2.34·73-s + 81-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 1.83·171-s + 0.0760·173-s − 6.04·175-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5859410580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5859410580\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.16761401018975576072128546912, −9.960644255468066515084109212125, −9.351770335484150192455132885692, −9.204770062142631447508928788500, −8.769787722509297637281614053890, −8.412089248664779769199288113932, −7.45556470841348436466321618004, −7.43590336353570720291134580689, −6.76865001662113966146381760546, −6.30212896485923721085145632104, −6.25752534180307998816673657423, −5.59665502603094405824843497903, −5.12799861132040569122094833053, −4.59908814655601506373740050463, −3.69832705049264906205962252807, −3.21295115422570410239520885970, −3.01966082202961335467321198093, −2.76664296321127246458074039600, −1.43059509108331842047903431619, −0.36983345952922080602492014411,
0.36983345952922080602492014411, 1.43059509108331842047903431619, 2.76664296321127246458074039600, 3.01966082202961335467321198093, 3.21295115422570410239520885970, 3.69832705049264906205962252807, 4.59908814655601506373740050463, 5.12799861132040569122094833053, 5.59665502603094405824843497903, 6.25752534180307998816673657423, 6.30212896485923721085145632104, 6.76865001662113966146381760546, 7.43590336353570720291134580689, 7.45556470841348436466321618004, 8.412089248664779769199288113932, 8.769787722509297637281614053890, 9.204770062142631447508928788500, 9.351770335484150192455132885692, 9.960644255468066515084109212125, 10.16761401018975576072128546912