| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s + 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s − 37-s − 38-s + 41-s + 43-s + 44-s − 46-s + 47-s + 49-s + 52-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s + 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s − 37-s − 38-s + 41-s + 43-s + 44-s − 46-s + 47-s + 49-s + 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.341134804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341134804\) |
| \(L(1)\) |
\(\approx\) |
\(0.9462476412\) |
| \(L(1)\) |
\(\approx\) |
\(0.9462476412\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 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
−21.31401386293400391169227158604, −20.70302466869491502452256176942, −20.15470531266170838810836586783, −19.06719514778323916756133936442, −18.55554570133165433831982303485, −17.72081737558710489650838404880, −17.0683772562797360338740774922, −16.34900798818961558641787781595, −15.47676986445680817362345322082, −14.59667559294592425327531676510, −13.99573850002404351862609148609, −12.646742790068118625133500005065, −11.76279330679558210747159360935, −11.15337546771998320613683298273, −10.47987930640269225548109694850, −9.202890962593827662039849591389, −8.94627410816928396166618458845, −7.73447162010634853999840187090, −7.30127319191092898670205389615, −6.08770961428783772621117327287, −5.35434340734945061024366618446, −3.94879418037131601125225809559, −3.03098123399731643973156392406, −1.61502481431745072112188342585, −1.127459248564943814493284574630,
1.127459248564943814493284574630, 1.61502481431745072112188342585, 3.03098123399731643973156392406, 3.94879418037131601125225809559, 5.35434340734945061024366618446, 6.08770961428783772621117327287, 7.30127319191092898670205389615, 7.73447162010634853999840187090, 8.94627410816928396166618458845, 9.202890962593827662039849591389, 10.47987930640269225548109694850, 11.15337546771998320613683298273, 11.76279330679558210747159360935, 12.646742790068118625133500005065, 13.99573850002404351862609148609, 14.59667559294592425327531676510, 15.47676986445680817362345322082, 16.34900798818961558641787781595, 17.0683772562797360338740774922, 17.72081737558710489650838404880, 18.55554570133165433831982303485, 19.06719514778323916756133936442, 20.15470531266170838810836586783, 20.70302466869491502452256176942, 21.31401386293400391169227158604
