| L(s) = 1 | − 2·4-s − 4·7-s + 4·16-s + 12·23-s − 25-s + 8·28-s + 12·29-s − 2·37-s − 8·43-s + 9·49-s − 12·53-s − 8·64-s − 8·67-s − 12·71-s − 8·79-s − 24·92-s + 2·100-s − 8·109-s − 16·112-s − 18·113-s − 24·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 16-s + 2.50·23-s − 1/5·25-s + 1.51·28-s + 2.22·29-s − 0.328·37-s − 1.21·43-s + 9/7·49-s − 1.64·53-s − 64-s − 0.977·67-s − 1.42·71-s − 0.900·79-s − 2.50·92-s + 1/5·100-s − 0.766·109-s − 1.51·112-s − 1.69·113-s − 2.22·116-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
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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
−7.58874714832343084329637604910, −6.93077463999897090620221680949, −6.78748557273936452939065635555, −6.37267959175031555681810711961, −5.82264431514856657329791225275, −5.39891934799349844025192851186, −4.81226764141306313980369427340, −4.60321092299125558019936944698, −4.02983934227065081286737166090, −3.27219609929124842463743759194, −3.08406070673207045211418059498, −2.75901500486628697286908180025, −1.55136531174937716397365757553, −0.884895111348097055490627453097, 0,
0.884895111348097055490627453097, 1.55136531174937716397365757553, 2.75901500486628697286908180025, 3.08406070673207045211418059498, 3.27219609929124842463743759194, 4.02983934227065081286737166090, 4.60321092299125558019936944698, 4.81226764141306313980369427340, 5.39891934799349844025192851186, 5.82264431514856657329791225275, 6.37267959175031555681810711961, 6.78748557273936452939065635555, 6.93077463999897090620221680949, 7.58874714832343084329637604910
