| L(s) = 1 | − 4·3-s + 2·5-s − 2·7-s + 6·9-s + 2·11-s + 2·13-s − 8·15-s − 2·17-s + 2·19-s + 8·21-s + 2·23-s + 3·25-s + 4·27-s + 6·29-s + 2·31-s − 8·33-s − 4·35-s − 10·37-s − 8·39-s − 8·41-s − 8·43-s + 12·45-s + 2·47-s + 3·49-s + 8·51-s − 14·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s − 0.755·7-s + 2·9-s + 0.603·11-s + 0.554·13-s − 2.06·15-s − 0.485·17-s + 0.458·19-s + 1.74·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s − 1.39·33-s − 0.676·35-s − 1.64·37-s − 1.28·39-s − 1.24·41-s − 1.21·43-s + 1.78·45-s + 0.291·47-s + 3/7·49-s + 1.12·51-s − 1.92·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163244705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163244705\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.186274701716392366393813704598, −8.053310770536038546101415107672, −7.04784912433877600333725961980, −6.79635309634068407321094202521, −6.71272381858003560925133403697, −6.62512623226233436129155415369, −5.99315221064332422531579841511, −5.83767784859191261404034025144, −5.35014471702454459723980171970, −5.25649386679768376749062680633, −4.68941108682314028770906250511, −4.65061072071387525111652100457, −3.72374592446915358912035296462, −3.61066040592259378630563523080, −2.88691340112206923875506277086, −2.69718163437861589422131314482, −1.76002392868902713310418394936, −1.52903130194778775408956197135, −0.72656273358875899614645617462, −0.48171704722400981211966423514,
0.48171704722400981211966423514, 0.72656273358875899614645617462, 1.52903130194778775408956197135, 1.76002392868902713310418394936, 2.69718163437861589422131314482, 2.88691340112206923875506277086, 3.61066040592259378630563523080, 3.72374592446915358912035296462, 4.65061072071387525111652100457, 4.68941108682314028770906250511, 5.25649386679768376749062680633, 5.35014471702454459723980171970, 5.83767784859191261404034025144, 5.99315221064332422531579841511, 6.62512623226233436129155415369, 6.71272381858003560925133403697, 6.79635309634068407321094202521, 7.04784912433877600333725961980, 8.053310770536038546101415107672, 8.186274701716392366393813704598
