| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s + 2·11-s − 2·13-s − 4·15-s + 2·19-s − 4·21-s − 4·23-s + 3·25-s + 4·27-s − 10·29-s − 8·31-s + 4·33-s + 4·35-s − 6·37-s − 4·39-s − 6·41-s − 2·43-s − 6·45-s − 4·47-s − 6·49-s − 8·53-s − 4·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.458·19-s − 0.872·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.986·37-s − 0.640·39-s − 0.937·41-s − 0.304·43-s − 0.894·45-s − 0.583·47-s − 6/7·49-s − 1.09·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 254 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.047781042799046620400310430892, −8.018700065643198320488718630837, −7.41124533938094127204662448612, −7.09169812989824302234904341846, −6.74308169863568657540517064202, −6.69628788248964729672411516190, −5.95317888954269062084566936028, −5.52410802075728692917584567547, −5.09305002972275259592193149520, −4.85521217763302722165632768723, −4.01476445622508235875270410743, −3.90265309451911877529315573453, −3.48464912125018259896618352741, −3.44037730192338821512068471497, −2.57943629223702597118139084738, −2.41097678466931265531289237094, −1.55571310248456849695728013619, −1.42343718162258186510076534025, 0, 0,
1.42343718162258186510076534025, 1.55571310248456849695728013619, 2.41097678466931265531289237094, 2.57943629223702597118139084738, 3.44037730192338821512068471497, 3.48464912125018259896618352741, 3.90265309451911877529315573453, 4.01476445622508235875270410743, 4.85521217763302722165632768723, 5.09305002972275259592193149520, 5.52410802075728692917584567547, 5.95317888954269062084566936028, 6.69628788248964729672411516190, 6.74308169863568657540517064202, 7.09169812989824302234904341846, 7.41124533938094127204662448612, 8.018700065643198320488718630837, 8.047781042799046620400310430892
