| L(s) = 1 | + 4-s − 5·9-s − 12·11-s + 16-s + 2·19-s − 5·25-s + 18·29-s − 8·31-s − 5·36-s − 12·44-s − 13·49-s + 18·59-s − 20·61-s + 64-s − 12·71-s + 2·76-s − 20·79-s + 16·81-s − 24·89-s + 60·99-s − 5·100-s + 36·101-s + 22·109-s + 18·116-s + 86·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 5/3·9-s − 3.61·11-s + 1/4·16-s + 0.458·19-s − 25-s + 3.34·29-s − 1.43·31-s − 5/6·36-s − 1.80·44-s − 1.85·49-s + 2.34·59-s − 2.56·61-s + 1/8·64-s − 1.42·71-s + 0.229·76-s − 2.25·79-s + 16/9·81-s − 2.54·89-s + 6.03·99-s − 1/2·100-s + 3.58·101-s + 2.10·109-s + 1.67·116-s + 7.81·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.09897479975324709020702728875, −10.00077510413278104543356920343, −8.684413720393242419277779815238, −8.596508490746687564678463005243, −7.82846290933361228167589994195, −7.73591843742988285704805588989, −6.97164965644133669332392215517, −5.93652577489482387138388578466, −5.81027585652647529822118508915, −5.08282473776364419277313134171, −4.69521261142767966957106985399, −3.04737002497535337211222281725, −2.99432139105097413830284496966, −2.23852341950432947206046912003, 0,
2.23852341950432947206046912003, 2.99432139105097413830284496966, 3.04737002497535337211222281725, 4.69521261142767966957106985399, 5.08282473776364419277313134171, 5.81027585652647529822118508915, 5.93652577489482387138388578466, 6.97164965644133669332392215517, 7.73591843742988285704805588989, 7.82846290933361228167589994195, 8.596508490746687564678463005243, 8.684413720393242419277779815238, 10.00077510413278104543356920343, 10.09897479975324709020702728875
