| L(s) = 1 | + 7-s − 5·11-s + 8·17-s − 2·19-s + 7·23-s − 3·29-s − 4·31-s − 37-s + 2·41-s − 3·43-s − 6·47-s + 49-s − 10·53-s + 4·59-s + 6·61-s − 13·67-s + 5·71-s − 6·73-s − 5·77-s + 13·79-s + 16·83-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.50·11-s + 1.94·17-s − 0.458·19-s + 1.45·23-s − 0.557·29-s − 0.718·31-s − 0.164·37-s + 0.312·41-s − 0.457·43-s − 0.875·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.768·61-s − 1.58·67-s + 0.593·71-s − 0.702·73-s − 0.569·77-s + 1.46·79-s + 1.75·83-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.99611031252755, −13.42109324604258, −12.95167521104960, −12.65394586734108, −12.08935483567716, −11.50628232858550, −10.98717098518078, −10.51967949556061, −10.22825934049508, −9.465393346170175, −9.170326902079378, −8.356020363643218, −7.903590192490567, −7.663343120881651, −7.031590439563322, −6.412199080774041, −5.666244576897042, −5.226941767345742, −4.994018414503920, −4.179662335469477, −3.334534459181108, −3.099302386982214, −2.305580191904200, −1.615236185956789, −0.8884804228841995, 0,
0.8884804228841995, 1.615236185956789, 2.305580191904200, 3.099302386982214, 3.334534459181108, 4.179662335469477, 4.994018414503920, 5.226941767345742, 5.666244576897042, 6.412199080774041, 7.031590439563322, 7.663343120881651, 7.903590192490567, 8.356020363643218, 9.170326902079378, 9.465393346170175, 10.22825934049508, 10.51967949556061, 10.98717098518078, 11.50628232858550, 12.08935483567716, 12.65394586734108, 12.95167521104960, 13.42109324604258, 13.99611031252755
