| L(s) = 1 | − 3-s + 4-s − 4·7-s + 9-s − 12-s − 4·13-s + 16-s + 4·21-s − 6·25-s − 27-s − 4·28-s + 16·31-s + 36-s + 4·39-s − 24·43-s − 48-s − 2·49-s − 4·52-s + 8·61-s − 4·63-s + 64-s − 24·67-s − 20·73-s + 6·75-s − 12·79-s + 81-s + 4·84-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.872·21-s − 6/5·25-s − 0.192·27-s − 0.755·28-s + 2.87·31-s + 1/6·36-s + 0.640·39-s − 3.65·43-s − 0.144·48-s − 2/7·49-s − 0.554·52-s + 1.02·61-s − 0.503·63-s + 1/8·64-s − 2.93·67-s − 2.34·73-s + 0.692·75-s − 1.35·79-s + 1/9·81-s + 0.436·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.832860358169758492996037846203, −9.630000082494789242014648422170, −8.623377758533207484865735274631, −8.259124288211168671734267271865, −7.48954786032446015166944686975, −7.04614505207963284764458800400, −6.43907907295841169043019574490, −6.25250569926844252670184221360, −5.55267882834237934594942614166, −4.78570582427944795062347753374, −4.27895780918103049245443906167, −3.16921099094622245450045571331, −2.91631229938310047782795220333, −1.72081981979020341015639926740, 0,
1.72081981979020341015639926740, 2.91631229938310047782795220333, 3.16921099094622245450045571331, 4.27895780918103049245443906167, 4.78570582427944795062347753374, 5.55267882834237934594942614166, 6.25250569926844252670184221360, 6.43907907295841169043019574490, 7.04614505207963284764458800400, 7.48954786032446015166944686975, 8.259124288211168671734267271865, 8.623377758533207484865735274631, 9.630000082494789242014648422170, 9.832860358169758492996037846203
