| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s + 3·7-s + 4·8-s + 2·10-s + 3·11-s + 3·12-s + 2·13-s + 6·14-s + 15-s + 5·16-s + 4·17-s + 3·20-s + 3·21-s + 6·22-s − 8·23-s + 4·24-s + 4·26-s − 27-s + 9·28-s − 10·29-s + 2·30-s + 11·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s + 1.41·8-s + 0.632·10-s + 0.904·11-s + 0.866·12-s + 0.554·13-s + 1.60·14-s + 0.258·15-s + 5/4·16-s + 0.970·17-s + 0.670·20-s + 0.654·21-s + 1.27·22-s − 1.66·23-s + 0.816·24-s + 0.784·26-s − 0.192·27-s + 1.70·28-s − 1.85·29-s + 0.365·30-s + 1.97·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.174095332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.174095332\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_2$ | \( 1 - 11 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.15490639733422592398993750236, −10.06313336648752984205782750850, −9.291772934936483037937818557628, −9.259186178159370207996579597284, −8.355019401360742023099072822766, −7.976943537951356950966702021411, −7.929063340094521089099421073307, −7.31338212734660411489206356737, −6.69896880412838919509821790879, −6.26054626222910571322493439985, −5.75902762572781259140433119921, −5.68280636162624683313065548242, −4.86889098220833586022263647804, −4.50015076232325173498030338145, −3.87030646276061031301297979428, −3.76954057489232609354877021191, −2.87811219297513495013808418306, −2.46838012742712737135329819441, −1.62380532765996854711342145857, −1.36320197114878326546102890200,
1.36320197114878326546102890200, 1.62380532765996854711342145857, 2.46838012742712737135329819441, 2.87811219297513495013808418306, 3.76954057489232609354877021191, 3.87030646276061031301297979428, 4.50015076232325173498030338145, 4.86889098220833586022263647804, 5.68280636162624683313065548242, 5.75902762572781259140433119921, 6.26054626222910571322493439985, 6.69896880412838919509821790879, 7.31338212734660411489206356737, 7.929063340094521089099421073307, 7.976943537951356950966702021411, 8.355019401360742023099072822766, 9.259186178159370207996579597284, 9.291772934936483037937818557628, 10.06313336648752984205782750850, 10.15490639733422592398993750236
