| L(s) = 1 | − 4-s + 7-s + 4·13-s + 16-s − 9·19-s − 6·25-s − 28-s − 16·31-s + 2·37-s + 3·43-s − 6·49-s − 4·52-s − 7·61-s − 64-s − 12·67-s − 14·73-s + 9·76-s + 4·91-s − 15·97-s + 6·100-s + 34·103-s + 28·109-s + 112-s + 121-s + 16·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.377·7-s + 1.10·13-s + 1/4·16-s − 2.06·19-s − 6/5·25-s − 0.188·28-s − 2.87·31-s + 0.328·37-s + 0.457·43-s − 6/7·49-s − 0.554·52-s − 0.896·61-s − 1/8·64-s − 1.46·67-s − 1.63·73-s + 1.03·76-s + 0.419·91-s − 1.52·97-s + 3/5·100-s + 3.35·103-s + 2.68·109-s + 0.0944·112-s + 1/11·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 13 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 136 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−8.830848494037951807101019661739, −8.720417440619100335970991907979, −8.062387213693486305946561160968, −7.55227181161853856703885648199, −7.19405103428111507643226937689, −6.30870657845361576381140527985, −6.01435376106891931503404507132, −5.60393979966852728518119703099, −4.78800566470201649955945266904, −4.29868099228912376084565120405, −3.81327193351115246312146514208, −3.24505475327997688747257426289, −2.11758128973097040807198938312, −1.59139990733728620447570002364, 0,
1.59139990733728620447570002364, 2.11758128973097040807198938312, 3.24505475327997688747257426289, 3.81327193351115246312146514208, 4.29868099228912376084565120405, 4.78800566470201649955945266904, 5.60393979966852728518119703099, 6.01435376106891931503404507132, 6.30870657845361576381140527985, 7.19405103428111507643226937689, 7.55227181161853856703885648199, 8.062387213693486305946561160968, 8.720417440619100335970991907979, 8.830848494037951807101019661739
