| L(s) = 1 | − 3-s + 4·5-s − 4·7-s + 3·9-s + 3·11-s − 10·13-s − 4·15-s + 17-s − 6·19-s + 4·21-s + 4·23-s + 5·25-s − 8·27-s + 16·29-s − 3·33-s − 16·35-s + 8·37-s + 10·39-s − 16·41-s + 20·43-s + 12·45-s + 2·47-s + 9·49-s − 51-s − 3·53-s + 12·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.51·7-s + 9-s + 0.904·11-s − 2.77·13-s − 1.03·15-s + 0.242·17-s − 1.37·19-s + 0.872·21-s + 0.834·23-s + 25-s − 1.53·27-s + 2.97·29-s − 0.522·33-s − 2.70·35-s + 1.31·37-s + 1.60·39-s − 2.49·41-s + 3.04·43-s + 1.78·45-s + 0.291·47-s + 9/7·49-s − 0.140·51-s − 0.412·53-s + 1.61·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516853620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516853620\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−11.14016967911275198332198693851, −10.35486219380780348092691761783, −10.30833381592883759392033763398, −9.804646222048664801357594556682, −9.739531373154519506040565257706, −9.135565609968013842425093580532, −8.939380906251546439201050092857, −7.939229705611444732851104076085, −7.38835861956298908950794173935, −6.83022545110749464947171567896, −6.61798013325581887009937250510, −6.12057786449719530044730100420, −5.82696301981483944496796448661, −4.92772260857320028913971866498, −4.74676173703891443300394719050, −4.05756286642614708584885173730, −3.10334143367717061002397666597, −2.41017011917080799956297620662, −2.06167778531779518806792174935, −0.77736879848003819330441892225,
0.77736879848003819330441892225, 2.06167778531779518806792174935, 2.41017011917080799956297620662, 3.10334143367717061002397666597, 4.05756286642614708584885173730, 4.74676173703891443300394719050, 4.92772260857320028913971866498, 5.82696301981483944496796448661, 6.12057786449719530044730100420, 6.61798013325581887009937250510, 6.83022545110749464947171567896, 7.38835861956298908950794173935, 7.939229705611444732851104076085, 8.939380906251546439201050092857, 9.135565609968013842425093580532, 9.739531373154519506040565257706, 9.804646222048664801357594556682, 10.30833381592883759392033763398, 10.35486219380780348092691761783, 11.14016967911275198332198693851
