| L(s) = 1 | − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 8·23-s + 6·29-s − 8·31-s + 6·37-s + 6·41-s − 4·43-s − 7·49-s + 2·53-s − 4·59-s + 2·61-s + 4·67-s + 8·71-s − 10·73-s + 8·79-s − 4·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s − 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.40723713140756, −15.86339438152485, −15.18368519819409, −14.77093452268043, −14.30234101084824, −13.47172081259227, −12.89260027578312, −12.68991299830959, −11.95036362627315, −11.05203856476603, −10.87622806389476, −10.12434474671448, −9.578990932597495, −8.903055293706380, −8.255796980767851, −7.677362533380482, −7.112325659768094, −6.423435442961949, −5.649879426782975, −5.010032767020927, −4.548653159077549, −3.537100377471948, −2.805684776949022, −2.218456936948766, −1.081774835208433, 0,
1.081774835208433, 2.218456936948766, 2.805684776949022, 3.537100377471948, 4.548653159077549, 5.010032767020927, 5.649879426782975, 6.423435442961949, 7.112325659768094, 7.677362533380482, 8.255796980767851, 8.903055293706380, 9.578990932597495, 10.12434474671448, 10.87622806389476, 11.05203856476603, 11.95036362627315, 12.68991299830959, 12.89260027578312, 13.47172081259227, 14.30234101084824, 14.77093452268043, 15.18368519819409, 15.86339438152485, 16.40723713140756
