| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.426574226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426574226\) |
| \(L(1)\) |
\(\approx\) |
\(1.219174394\) |
| \(L(1)\) |
\(\approx\) |
\(1.219174394\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 83 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.7740995421901859192081173228, −21.62679627194695333443342961583, −20.936196562259149381446212084, −20.74151289466099771480197293994, −18.90273586087853229047979049470, −18.29012396191195618004372684357, −17.72401033203860753224937399579, −16.96895508981321198987612945097, −16.13535823918146349277061252392, −15.253576697845536504656788894282, −14.14181886733345952946389848413, −13.38429032723916218707511015021, −12.56926844991406536388321506934, −11.51712323384679781054405406486, −10.797592306226484300697869103588, −10.1355262777652321752784913734, −9.1081758417829043482873129001, −7.916504638350049121326854193730, −7.06004599684196330146172488494, −5.75944615737523917020031999341, −5.462077450735719058500255357835, −4.50921627787091457746011965130, −3.02880733763852492788508882948, −1.65096963303009147087346008242, −0.91983156285504112405518249016,
0.91983156285504112405518249016, 1.65096963303009147087346008242, 3.02880733763852492788508882948, 4.50921627787091457746011965130, 5.462077450735719058500255357835, 5.75944615737523917020031999341, 7.06004599684196330146172488494, 7.916504638350049121326854193730, 9.1081758417829043482873129001, 10.1355262777652321752784913734, 10.797592306226484300697869103588, 11.51712323384679781054405406486, 12.56926844991406536388321506934, 13.38429032723916218707511015021, 14.14181886733345952946389848413, 15.253576697845536504656788894282, 16.13535823918146349277061252392, 16.96895508981321198987612945097, 17.72401033203860753224937399579, 18.29012396191195618004372684357, 18.90273586087853229047979049470, 20.74151289466099771480197293994, 20.936196562259149381446212084, 21.62679627194695333443342961583, 22.7740995421901859192081173228
