| L(s) = 1 | + 2·3-s − 4·11-s − 2·13-s − 8·17-s − 2·19-s + 4·23-s − 2·27-s − 8·31-s − 8·33-s + 2·37-s − 4·39-s + 8·41-s + 8·43-s + 8·47-s − 14·49-s − 16·51-s + 6·53-s − 4·57-s + 4·59-s − 20·61-s + 2·67-s + 8·69-s − 8·71-s − 4·79-s − 81-s − 4·83-s − 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.20·11-s − 0.554·13-s − 1.94·17-s − 0.458·19-s + 0.834·23-s − 0.384·27-s − 1.43·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.24·41-s + 1.21·43-s + 1.16·47-s − 2·49-s − 2.24·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 2.56·61-s + 0.244·67-s + 0.963·69-s − 0.949·71-s − 0.450·79-s − 1/9·81-s − 0.439·83-s − 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 158 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 168 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.65223810750271797473850552530, −7.58794913059753531594450102048, −7.03628878599344396962049212370, −6.82133268616824868993829131105, −6.32431614742148721350237626342, −6.02326608542179201285357404961, −5.49265994879088240327071448344, −5.31486391899852593254860848236, −4.77299901506690836393956291229, −4.48043453858193683983571758970, −4.08291082724741998012469370705, −3.79015832423466393247169031757, −3.08896114750917476695646062309, −2.79054634239046173645032932582, −2.57540594359899689112692815756, −2.24320743601073332176422190687, −1.74317005923362344676981041672, −1.08602438645589400797360644082, 0, 0,
1.08602438645589400797360644082, 1.74317005923362344676981041672, 2.24320743601073332176422190687, 2.57540594359899689112692815756, 2.79054634239046173645032932582, 3.08896114750917476695646062309, 3.79015832423466393247169031757, 4.08291082724741998012469370705, 4.48043453858193683983571758970, 4.77299901506690836393956291229, 5.31486391899852593254860848236, 5.49265994879088240327071448344, 6.02326608542179201285357404961, 6.32431614742148721350237626342, 6.82133268616824868993829131105, 7.03628878599344396962049212370, 7.58794913059753531594450102048, 7.65223810750271797473850552530
