| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 3·5-s − 2·6-s + 4·7-s − 2·9-s − 6·10-s − 2·11-s + 2·12-s − 7·13-s − 8·14-s + 3·15-s − 4·16-s + 2·17-s + 4·18-s − 6·19-s + 6·20-s + 4·21-s + 4·22-s + 2·23-s + 4·25-s + 14·26-s − 5·27-s + 8·28-s − 6·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 1.34·5-s − 0.816·6-s + 1.51·7-s − 2/3·9-s − 1.89·10-s − 0.603·11-s + 0.577·12-s − 1.94·13-s − 2.13·14-s + 0.774·15-s − 16-s + 0.485·17-s + 0.942·18-s − 1.37·19-s + 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s + 4/5·25-s + 2.74·26-s − 0.962·27-s + 1.51·28-s − 1.09·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4021 | \( 1+O(T) \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.191512266729550826988751605117, −7.66163751240228221106644028436, −7.02775442295137546944896136210, −5.85312347802252794461035828363, −5.14908600587112591673920829270, −4.49262689425840127793219031456, −2.75817315801570221960210431939, −2.14192786388500324377828550891, −1.61585751454237153489456050181, 0,
1.61585751454237153489456050181, 2.14192786388500324377828550891, 2.75817315801570221960210431939, 4.49262689425840127793219031456, 5.14908600587112591673920829270, 5.85312347802252794461035828363, 7.02775442295137546944896136210, 7.66163751240228221106644028436, 8.191512266729550826988751605117
