| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s + 2·11-s − 10·13-s + 4·15-s − 12·17-s + 2·19-s − 4·21-s + 2·23-s + 3·25-s + 4·27-s + 4·29-s + 2·31-s + 4·33-s − 4·35-s − 2·37-s − 20·39-s − 12·41-s + 6·45-s + 2·47-s + 49-s − 24·51-s − 4·53-s + 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s − 2.77·13-s + 1.03·15-s − 2.91·17-s + 0.458·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s + 0.742·29-s + 0.359·31-s + 0.696·33-s − 0.676·35-s − 0.328·37-s − 3.20·39-s − 1.87·41-s + 0.894·45-s + 0.291·47-s + 1/7·49-s − 3.36·51-s − 0.549·53-s + 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 95 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 200 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 26 T + 312 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−7.965430315481373096311694697342, −7.56563281762820846647998368989, −7.06981355453028434516610350409, −7.04431874408597642669331231541, −6.62252933909551042072847675667, −6.43986158985450803385820514785, −5.89381691974811526138813245647, −5.35333211291605889158181486220, −4.92007561294516203451051555786, −4.67396457819577023345918723150, −4.26880892501509305639341684560, −4.06079973146719300289147148627, −3.09510181951742664111339256701, −2.99675759979789735180506887140, −2.57037988150066928917858362030, −2.34727380431192549136597441584, −1.65658914079780922967516120488, −1.44637368836418714828081992504, 0, 0,
1.44637368836418714828081992504, 1.65658914079780922967516120488, 2.34727380431192549136597441584, 2.57037988150066928917858362030, 2.99675759979789735180506887140, 3.09510181951742664111339256701, 4.06079973146719300289147148627, 4.26880892501509305639341684560, 4.67396457819577023345918723150, 4.92007561294516203451051555786, 5.35333211291605889158181486220, 5.89381691974811526138813245647, 6.43986158985450803385820514785, 6.62252933909551042072847675667, 7.04431874408597642669331231541, 7.06981355453028434516610350409, 7.56563281762820846647998368989, 7.965430315481373096311694697342
