| L(s) = 1 | − 2·4-s − 6·7-s − 2·13-s + 19-s − 25-s + 12·28-s + 3·31-s + 2·37-s + 8·43-s + 14·49-s + 4·52-s − 14·61-s + 8·64-s + 8·67-s − 14·73-s − 2·76-s + 12·79-s + 12·91-s + 28·97-s + 2·100-s − 6·103-s + 7·109-s − 3·121-s − 6·124-s + 127-s + 131-s − 6·133-s + ⋯ |
| L(s) = 1 | − 4-s − 2.26·7-s − 0.554·13-s + 0.229·19-s − 1/5·25-s + 2.26·28-s + 0.538·31-s + 0.328·37-s + 1.21·43-s + 2·49-s + 0.554·52-s − 1.79·61-s + 64-s + 0.977·67-s − 1.63·73-s − 0.229·76-s + 1.35·79-s + 1.25·91-s + 2.84·97-s + 1/5·100-s − 0.591·103-s + 0.670·109-s − 0.272·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
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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
−8.080451873692148054035255659032, −7.60483619511905417144497561702, −7.17046548460249144690289114907, −6.64868797335954987802648606278, −6.27458987075753739754017400329, −5.90703069937853517870497828523, −5.34919797396805925423634387050, −4.71147954673986466796135546198, −4.34595579050750539330271265221, −3.74395923694660727602660911129, −3.23602205336714547399982781168, −2.82803093830948032171434194362, −2.09593589379171299463948610753, −0.78742035018376813173338161309, 0,
0.78742035018376813173338161309, 2.09593589379171299463948610753, 2.82803093830948032171434194362, 3.23602205336714547399982781168, 3.74395923694660727602660911129, 4.34595579050750539330271265221, 4.71147954673986466796135546198, 5.34919797396805925423634387050, 5.90703069937853517870497828523, 6.27458987075753739754017400329, 6.64868797335954987802648606278, 7.17046548460249144690289114907, 7.60483619511905417144497561702, 8.080451873692148054035255659032
