| L(s) = 1 | + 3-s − 3·4-s + 2·5-s + 7-s + 9-s − 3·12-s + 2·15-s + 5·16-s + 4·17-s − 6·20-s + 21-s + 3·25-s + 27-s − 3·28-s + 2·35-s − 3·36-s − 4·37-s − 12·41-s + 8·43-s + 2·45-s + 16·47-s + 5·48-s + 49-s + 4·51-s + 8·59-s − 6·60-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.866·12-s + 0.516·15-s + 5/4·16-s + 0.970·17-s − 1.34·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s − 0.566·28-s + 0.338·35-s − 1/2·36-s − 0.657·37-s − 1.87·41-s + 1.21·43-s + 0.298·45-s + 2.33·47-s + 0.721·48-s + 1/7·49-s + 0.560·51-s + 1.04·59-s − 0.774·60-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875587480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875587480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.158915116725801964521352538031, −8.447489066361171687346291758504, −8.275102727179600443330310060644, −7.82733728798443659692174656726, −6.97944382340517249810347443111, −6.82600380115164878858049675491, −5.81605182603986018580602603999, −5.37573342552736399524296425305, −5.25448844749250196408410433194, −4.33309058948755389799390879771, −4.05504421439464721652879145719, −3.34833988846054044432199186298, −2.63294756479994443235170715895, −1.79849192746842450415563444508, −0.888154170390839902230274080915,
0.888154170390839902230274080915, 1.79849192746842450415563444508, 2.63294756479994443235170715895, 3.34833988846054044432199186298, 4.05504421439464721652879145719, 4.33309058948755389799390879771, 5.25448844749250196408410433194, 5.37573342552736399524296425305, 5.81605182603986018580602603999, 6.82600380115164878858049675491, 6.97944382340517249810347443111, 7.82733728798443659692174656726, 8.275102727179600443330310060644, 8.447489066361171687346291758504, 9.158915116725801964521352538031
