| L(s) = 1 | − 2·4-s − 7-s + 9-s + 4·16-s − 6·17-s − 3·23-s + 25-s + 2·28-s − 2·31-s − 2·36-s − 9·41-s + 6·47-s − 6·49-s − 63-s − 8·64-s + 12·68-s − 18·71-s − 14·73-s + 16·79-s − 8·81-s + 18·89-s + 6·92-s + 10·97-s − 2·100-s − 20·103-s − 4·112-s − 12·113-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 1/3·9-s + 16-s − 1.45·17-s − 0.625·23-s + 1/5·25-s + 0.377·28-s − 0.359·31-s − 1/3·36-s − 1.40·41-s + 0.875·47-s − 6/7·49-s − 0.125·63-s − 64-s + 1.45·68-s − 2.13·71-s − 1.63·73-s + 1.80·79-s − 8/9·81-s + 1.90·89-s + 0.625·92-s + 1.01·97-s − 1/5·100-s − 1.97·103-s − 0.377·112-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.359332075791933937211567389414, −8.957968275564379316972193692586, −8.761894786500716272048485510380, −7.972677684993947820291342050002, −7.62871338866164943105134050817, −6.85314903204422003161681462899, −6.45486350467632186705085176932, −5.82101472756553329149991274450, −5.15918987614405963381559112272, −4.61246608963063032013691796294, −4.07948123630324092683197955674, −3.49209230240659697440629072958, −2.61516607397066105724389825559, −1.56342537955818974467491020682, 0,
1.56342537955818974467491020682, 2.61516607397066105724389825559, 3.49209230240659697440629072958, 4.07948123630324092683197955674, 4.61246608963063032013691796294, 5.15918987614405963381559112272, 5.82101472756553329149991274450, 6.45486350467632186705085176932, 6.85314903204422003161681462899, 7.62871338866164943105134050817, 7.972677684993947820291342050002, 8.761894786500716272048485510380, 8.957968275564379316972193692586, 9.359332075791933937211567389414
