| L(s) = 1 | − 2·7-s + 11-s − 2·13-s − 2·19-s − 23-s − 5·25-s − 10·29-s + 4·31-s − 2·37-s + 2·41-s − 2·43-s + 8·47-s − 3·49-s − 4·53-s − 12·59-s + 6·61-s − 2·67-s − 6·73-s − 2·77-s + 2·79-s + 4·83-s + 8·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 0.554·13-s − 0.458·19-s − 0.208·23-s − 25-s − 1.85·29-s + 0.718·31-s − 0.328·37-s + 0.312·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.549·53-s − 1.56·59-s + 0.768·61-s − 0.244·67-s − 0.702·73-s − 0.227·77-s + 0.225·79-s + 0.439·83-s + 0.847·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.65001564672371, −13.05026058950316, −12.71468422998490, −12.17860131874782, −11.77534920276533, −11.24361868964333, −10.71000633598771, −10.22904490149445, −9.643557375894629, −9.387760638539038, −8.908867993768972, −8.190683633778777, −7.767922967116773, −7.199545016900990, −6.795673553817521, −6.037677873208487, −5.889254780203816, −5.166485305421604, −4.464639821721586, −4.051346141629442, −3.388529600313430, −2.953945648478058, −2.073756584339516, −1.765388056054820, −0.6706637066426683, 0,
0.6706637066426683, 1.765388056054820, 2.073756584339516, 2.953945648478058, 3.388529600313430, 4.051346141629442, 4.464639821721586, 5.166485305421604, 5.889254780203816, 6.037677873208487, 6.795673553817521, 7.199545016900990, 7.767922967116773, 8.190683633778777, 8.908867993768972, 9.387760638539038, 9.643557375894629, 10.22904490149445, 10.71000633598771, 11.24361868964333, 11.77534920276533, 12.17860131874782, 12.71468422998490, 13.05026058950316, 13.65001564672371
