| L(s) = 1 | + 7-s − 3·9-s − 4·11-s − 6·13-s + 2·17-s − 4·19-s − 23-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 12·43-s − 12·47-s + 49-s + 10·53-s + 2·61-s − 3·63-s + 12·67-s − 8·71-s + 14·73-s − 4·77-s − 8·79-s + 9·81-s − 4·83-s + 6·89-s − 6·91-s + 10·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.82·43-s − 1.75·47-s + 1/7·49-s + 1.37·53-s + 0.256·61-s − 0.377·63-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.455·77-s − 0.900·79-s + 81-s − 0.439·83-s + 0.635·89-s − 0.628·91-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.50389226871077, −14.15188921666803, −13.41176504022586, −13.01032132490409, −12.39544369683273, −12.02909760670706, −11.46735967137142, −10.93513433244147, −10.44730847848391, −9.874835293237082, −9.549488106865449, −8.625142984552253, −8.375742142741619, −7.722034781453654, −7.412656091223387, −6.659795091448737, −6.010439653738106, −5.406069828135540, −5.022100534290571, −4.498660119147399, −3.683611508519223, −2.892740435684032, −2.429859116377826, −1.974887118450823, −0.7112500429186889, 0,
0.7112500429186889, 1.974887118450823, 2.429859116377826, 2.892740435684032, 3.683611508519223, 4.498660119147399, 5.022100534290571, 5.406069828135540, 6.010439653738106, 6.659795091448737, 7.412656091223387, 7.722034781453654, 8.375742142741619, 8.625142984552253, 9.549488106865449, 9.874835293237082, 10.44730847848391, 10.93513433244147, 11.46735967137142, 12.02909760670706, 12.39544369683273, 13.01032132490409, 13.41176504022586, 14.15188921666803, 14.50389226871077
