| L(s) = 1 | − 3·5-s + 3·11-s − 13-s + 12·17-s + 8·19-s − 3·23-s + 5·25-s + 3·29-s + 5·31-s + 4·37-s − 3·41-s + 43-s + 9·47-s + 12·53-s − 9·55-s + 3·59-s − 13·61-s + 3·65-s + 7·67-s + 24·71-s + 20·73-s − 11·79-s + 9·83-s − 36·85-s + 12·89-s − 24·95-s + 11·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s − 0.277·13-s + 2.91·17-s + 1.83·19-s − 0.625·23-s + 25-s + 0.557·29-s + 0.898·31-s + 0.657·37-s − 0.468·41-s + 0.152·43-s + 1.31·47-s + 1.64·53-s − 1.21·55-s + 0.390·59-s − 1.66·61-s + 0.372·65-s + 0.855·67-s + 2.84·71-s + 2.34·73-s − 1.23·79-s + 0.987·83-s − 3.90·85-s + 1.27·89-s − 2.46·95-s + 1.11·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.319291784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.319291784\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.105890648688986281629505052324, −7.932039095754783341348644935743, −7.79541603452542223006465933987, −7.46253276401390736968909661942, −6.99457691377863456282227520066, −6.57549982340142569083174315161, −6.42536399995803981029179137949, −5.56182083340702673658634069811, −5.52876379069087807078023722543, −5.23274886483859122615383116607, −4.67577952092098257811519411321, −4.20637727672175029854184478402, −3.72159708044528957798997012691, −3.62502513105806412459837991532, −3.24035441898331321716179609359, −2.69756767146503763468751867304, −2.28115010444354823183467351754, −1.23103351585545780779465793219, −1.08333234808291620716377415740, −0.62607065710655954729249065817,
0.62607065710655954729249065817, 1.08333234808291620716377415740, 1.23103351585545780779465793219, 2.28115010444354823183467351754, 2.69756767146503763468751867304, 3.24035441898331321716179609359, 3.62502513105806412459837991532, 3.72159708044528957798997012691, 4.20637727672175029854184478402, 4.67577952092098257811519411321, 5.23274886483859122615383116607, 5.52876379069087807078023722543, 5.56182083340702673658634069811, 6.42536399995803981029179137949, 6.57549982340142569083174315161, 6.99457691377863456282227520066, 7.46253276401390736968909661942, 7.79541603452542223006465933987, 7.932039095754783341348644935743, 8.105890648688986281629505052324
