| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 4·13-s + 16-s − 4·17-s − 12·23-s + 6·25-s + 4·27-s − 3·36-s + 8·39-s + 8·43-s + 2·48-s − 49-s − 8·51-s − 4·52-s + 8·53-s + 24·61-s − 64-s + 4·68-s − 24·69-s + 12·75-s + 5·81-s + 12·92-s − 6·100-s + 4·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.970·17-s − 2.50·23-s + 6/5·25-s + 0.769·27-s − 1/2·36-s + 1.28·39-s + 1.21·43-s + 0.288·48-s − 1/7·49-s − 1.12·51-s − 0.554·52-s + 1.09·53-s + 3.07·61-s − 1/8·64-s + 0.485·68-s − 2.88·69-s + 1.38·75-s + 5/9·81-s + 1.25·92-s − 3/5·100-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.450700732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.450700732\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.98284679500704824209555115162, −10.43906599472386037674039697815, −10.06788693215347469742102062536, −9.659921115453137208029938945955, −9.189931348597325525389078539261, −8.691171084522185913450514798227, −8.369919440255692162891186696016, −8.235604395395760847355183186408, −7.51287583689047377352540681663, −7.02718194012533614405126100233, −6.57885924456970419944274634727, −5.87789916850495982186630237276, −5.62467068425307463596678292367, −4.56248524829342132890619276028, −4.40700276552644831119540956017, −3.66579191453345194167135106980, −3.43035668030050780103166388735, −2.36850151416646713137635551837, −2.05695015814643121629296445367, −0.893805411861998558286043851328,
0.893805411861998558286043851328, 2.05695015814643121629296445367, 2.36850151416646713137635551837, 3.43035668030050780103166388735, 3.66579191453345194167135106980, 4.40700276552644831119540956017, 4.56248524829342132890619276028, 5.62467068425307463596678292367, 5.87789916850495982186630237276, 6.57885924456970419944274634727, 7.02718194012533614405126100233, 7.51287583689047377352540681663, 8.235604395395760847355183186408, 8.369919440255692162891186696016, 8.691171084522185913450514798227, 9.189931348597325525389078539261, 9.659921115453137208029938945955, 10.06788693215347469742102062536, 10.43906599472386037674039697815, 10.98284679500704824209555115162
