| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.020770485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.020770485\) |
| \(L(1)\) |
\(\approx\) |
\(3.078444144\) |
| \(L(1)\) |
\(\approx\) |
\(3.078444144\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1009 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 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
−21.586764735667207452441638406489, −21.06940735134846305265486244798, −20.20200799559455673975014248791, −19.73514190754304841276421905551, −18.48145236446189892950064803352, −17.79219381973837673804051815983, −16.8394662749852523856119408116, −15.79044975470555171831473797859, −15.02522826707610708990064215238, −14.3937078855760015527803322302, −13.86546933915786398736721066171, −12.98842167582520483970370098918, −12.53351811513257716674291744806, −11.200984252628400328617860738330, −10.43844239786348731307155723272, −9.68080096237761458954912772982, −8.477842438315604058613999921084, −7.77567557647173390131646430777, −6.8659123882384475833666825914, −5.88466456277595481702953605764, −4.79151045245996290094667967537, −4.41043898067921329817708079348, −2.93700499647063428954941259871, −2.23622050791856987123726437858, −1.7195094324241965875529115223,
1.7195094324241965875529115223, 2.23622050791856987123726437858, 2.93700499647063428954941259871, 4.41043898067921329817708079348, 4.79151045245996290094667967537, 5.88466456277595481702953605764, 6.8659123882384475833666825914, 7.77567557647173390131646430777, 8.477842438315604058613999921084, 9.68080096237761458954912772982, 10.43844239786348731307155723272, 11.200984252628400328617860738330, 12.53351811513257716674291744806, 12.98842167582520483970370098918, 13.86546933915786398736721066171, 14.3937078855760015527803322302, 15.02522826707610708990064215238, 15.79044975470555171831473797859, 16.8394662749852523856119408116, 17.79219381973837673804051815983, 18.48145236446189892950064803352, 19.73514190754304841276421905551, 20.20200799559455673975014248791, 21.06940735134846305265486244798, 21.586764735667207452441638406489
