| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s − 4·8-s + 8·10-s − 2·11-s + 2·13-s + 5·16-s − 4·17-s + 4·19-s − 12·20-s + 4·22-s + 4·23-s + 4·25-s − 4·26-s + 6·29-s + 8·31-s − 6·32-s + 8·34-s − 16·37-s − 8·38-s + 16·40-s − 8·41-s − 6·44-s − 8·46-s − 4·47-s − 8·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2.52·10-s − 0.603·11-s + 0.554·13-s + 5/4·16-s − 0.970·17-s + 0.917·19-s − 2.68·20-s + 0.852·22-s + 0.834·23-s + 4/5·25-s − 0.784·26-s + 1.11·29-s + 1.43·31-s − 1.06·32-s + 1.37·34-s − 2.63·37-s − 1.29·38-s + 2.52·40-s − 1.24·41-s − 0.904·44-s − 1.17·46-s − 0.583·47-s − 1.13·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 195 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 187 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.46049137598760973334234103539, −7.40796069000006079389694469817, −6.91385223220425808081746944684, −6.75233962065937090664584337379, −6.25908316740083590549954493843, −6.12334159214139217521822155353, −5.34213998835673534034361167885, −5.14125701725927132595083009572, −4.65561783200969301874243971212, −4.49084747260429885767686706042, −3.83310788664173361698306114070, −3.34700725314762400136380792892, −3.31284023661381886819398487300, −2.91039855330944351685260804831, −2.22328179050626222135070273776, −1.90632174822418623153702264420, −1.19889274395475451031651852341, −0.871112042738333590705794153600, 0, 0,
0.871112042738333590705794153600, 1.19889274395475451031651852341, 1.90632174822418623153702264420, 2.22328179050626222135070273776, 2.91039855330944351685260804831, 3.31284023661381886819398487300, 3.34700725314762400136380792892, 3.83310788664173361698306114070, 4.49084747260429885767686706042, 4.65561783200969301874243971212, 5.14125701725927132595083009572, 5.34213998835673534034361167885, 6.12334159214139217521822155353, 6.25908316740083590549954493843, 6.75233962065937090664584337379, 6.91385223220425808081746944684, 7.40796069000006079389694469817, 7.46049137598760973334234103539
