| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·5-s + 2·6-s − 2·9-s − 4·10-s − 2·12-s − 2·15-s − 4·16-s + 4·18-s + 4·20-s − 2·23-s − 7·25-s + 5·27-s + 4·30-s + 8·32-s − 4·36-s − 12·43-s − 4·45-s + 4·46-s + 16·47-s + 4·48-s − 10·49-s + 14·50-s − 12·53-s − 10·54-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s + 0.816·6-s − 2/3·9-s − 1.26·10-s − 0.577·12-s − 0.516·15-s − 16-s + 0.942·18-s + 0.894·20-s − 0.417·23-s − 7/5·25-s + 0.962·27-s + 0.730·30-s + 1.41·32-s − 2/3·36-s − 1.82·43-s − 0.596·45-s + 0.589·46-s + 2.33·47-s + 0.577·48-s − 1.42·49-s + 1.97·50-s − 1.64·53-s − 1.36·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.537544181722955003467047355519, −9.258074328098441277619654960607, −8.603539619290756001226038948684, −8.205136415893702144081639963978, −7.69242913833772582185010686143, −7.14892368575311887908494718560, −6.36261389471308870138602900888, −6.14904891822279137731207097129, −5.49252319411018247940447627282, −4.89830792778500137217036432202, −4.12475143940327773662034884992, −3.10894936791928072023408414397, −2.20758810273071278586475611416, −1.48884771837158153348689187504, 0,
1.48884771837158153348689187504, 2.20758810273071278586475611416, 3.10894936791928072023408414397, 4.12475143940327773662034884992, 4.89830792778500137217036432202, 5.49252319411018247940447627282, 6.14904891822279137731207097129, 6.36261389471308870138602900888, 7.14892368575311887908494718560, 7.69242913833772582185010686143, 8.205136415893702144081639963978, 8.603539619290756001226038948684, 9.258074328098441277619654960607, 9.537544181722955003467047355519
