| L(s) = 1 | + 4-s + 3·5-s − 3·7-s + 5·9-s − 6·13-s + 16-s + 3·20-s + 4·25-s − 3·28-s − 9·35-s + 5·36-s + 15·45-s + 6·47-s + 2·49-s − 6·52-s − 15·63-s + 64-s − 18·65-s − 12·73-s + 20·79-s + 3·80-s + 16·81-s + 12·83-s + 18·91-s + 24·97-s + 4·100-s − 3·112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s − 1.13·7-s + 5/3·9-s − 1.66·13-s + 1/4·16-s + 0.670·20-s + 4/5·25-s − 0.566·28-s − 1.52·35-s + 5/6·36-s + 2.23·45-s + 0.875·47-s + 2/7·49-s − 0.832·52-s − 1.88·63-s + 1/8·64-s − 2.23·65-s − 1.40·73-s + 2.25·79-s + 0.335·80-s + 16/9·81-s + 1.31·83-s + 1.88·91-s + 2.43·97-s + 2/5·100-s − 0.283·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.776640194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776640194\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−8.044232823266163812238795677911, −7.63482311406937681581586681382, −7.20670188067571374907984565476, −6.91567390174432327305544647075, −6.45871850949150718960282886455, −6.11468033463217861943153087036, −5.60007585681753022917158026351, −4.99805789276138837694498055970, −4.66504850751378008069282846265, −4.01102893203527148482379160677, −3.36933291908444446853727413458, −2.78542304823223274645134215750, −2.12357001650370786792821988508, −1.84305665235828701509630587478, −0.78779813422034744796473924681,
0.78779813422034744796473924681, 1.84305665235828701509630587478, 2.12357001650370786792821988508, 2.78542304823223274645134215750, 3.36933291908444446853727413458, 4.01102893203527148482379160677, 4.66504850751378008069282846265, 4.99805789276138837694498055970, 5.60007585681753022917158026351, 6.11468033463217861943153087036, 6.45871850949150718960282886455, 6.91567390174432327305544647075, 7.20670188067571374907984565476, 7.63482311406937681581586681382, 8.044232823266163812238795677911
