| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 2·12-s + 4·16-s + 14·19-s + 25-s − 5·27-s + 4·36-s + 4·43-s + 4·48-s + 13·49-s + 14·57-s − 8·64-s + 8·67-s + 12·73-s + 75-s − 28·76-s + 81-s + 20·97-s − 2·100-s + 10·108-s + 121-s + 127-s + 4·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 0.577·12-s + 16-s + 3.21·19-s + 1/5·25-s − 0.962·27-s + 2/3·36-s + 0.609·43-s + 0.577·48-s + 13/7·49-s + 1.85·57-s − 64-s + 0.977·67-s + 1.40·73-s + 0.115·75-s − 3.21·76-s + 1/9·81-s + 2.03·97-s − 1/5·100-s + 0.962·108-s + 1/11·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.168337928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168337928\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.78286852274440434328827933693, −7.58076572848560847226046710062, −7.19553572911651493423353544624, −6.55876208947319576080155133155, −5.91549977381981846056557469642, −5.59105829485694800308451145540, −5.15896630668513026297773949293, −4.93893101294987454153996327431, −4.15411621923531173204555847932, −3.64065759000290472724136420420, −3.36241881302723601435495391242, −2.81575756717332478539007541855, −2.25354744635648476764703055461, −1.22575280261751046715884988270, −0.69134678158912166368349150873,
0.69134678158912166368349150873, 1.22575280261751046715884988270, 2.25354744635648476764703055461, 2.81575756717332478539007541855, 3.36241881302723601435495391242, 3.64065759000290472724136420420, 4.15411621923531173204555847932, 4.93893101294987454153996327431, 5.15896630668513026297773949293, 5.59105829485694800308451145540, 5.91549977381981846056557469642, 6.55876208947319576080155133155, 7.19553572911651493423353544624, 7.58076572848560847226046710062, 7.78286852274440434328827933693
