| L(s) = 1 | − 2·3-s + 2·5-s − 3·7-s + 3·9-s − 5·11-s − 13-s − 4·15-s + 7·17-s + 5·19-s + 6·21-s − 23-s + 3·25-s − 4·27-s − 3·29-s + 3·31-s + 10·33-s − 6·35-s − 37-s + 2·39-s + 5·41-s − 8·43-s + 6·45-s − 3·47-s + 7·49-s − 14·51-s + 12·53-s − 10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.13·7-s + 9-s − 1.50·11-s − 0.277·13-s − 1.03·15-s + 1.69·17-s + 1.14·19-s + 1.30·21-s − 0.208·23-s + 3/5·25-s − 0.769·27-s − 0.557·29-s + 0.538·31-s + 1.74·33-s − 1.01·35-s − 0.164·37-s + 0.320·39-s + 0.780·41-s − 1.21·43-s + 0.894·45-s − 0.437·47-s + 49-s − 1.96·51-s + 1.64·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7931277100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7931277100\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 67 | $C_2$ | \( 1 + 16 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.834386875631270470147337739247, −7.965222810914602199925083533824, −7.962521081336394742696425201352, −7.34652946230400708963861224583, −7.21644469889368692083663739895, −6.72789934861155483865984621759, −6.39538521703792654825631984145, −5.75776499627003960135777796725, −5.71746453793740099762452193787, −5.35956540066640591085463062959, −5.27014477016054648434140693288, −4.40778082322961837370031961464, −4.33677419859553142863706998429, −3.33657558080948007539995983905, −3.29975950529957413986784776491, −2.72401406548918202179532631168, −2.32393136671671121357694060603, −1.50067273237838381261711393050, −1.10851823273325541150960932208, −0.30606750883466963191871585572,
0.30606750883466963191871585572, 1.10851823273325541150960932208, 1.50067273237838381261711393050, 2.32393136671671121357694060603, 2.72401406548918202179532631168, 3.29975950529957413986784776491, 3.33657558080948007539995983905, 4.33677419859553142863706998429, 4.40778082322961837370031961464, 5.27014477016054648434140693288, 5.35956540066640591085463062959, 5.71746453793740099762452193787, 5.75776499627003960135777796725, 6.39538521703792654825631984145, 6.72789934861155483865984621759, 7.21644469889368692083663739895, 7.34652946230400708963861224583, 7.962521081336394742696425201352, 7.965222810914602199925083533824, 8.834386875631270470147337739247
