| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1777 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1777 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.714107683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.714107683\) |
| \(L(1)\) |
\(\approx\) |
\(2.259211658\) |
| \(L(1)\) |
\(\approx\) |
\(2.259211658\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1777 | \( 1 \) |
| 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
−20.27405781989555772171566192103, −19.64124649610267716416998771591, −18.87329144144293239585017466762, −18.60501381875913101633718839974, −16.80753324942461502380856885938, −16.23447296547740273319079466184, −15.46565119145758220462093574284, −15.195521258546149673767869586061, −14.31830449861400584737582296853, −13.34330175417570285960751075531, −12.95541582369924354887084472964, −12.37268798493004718145995272131, −11.283886108127963377857183441934, −10.55797026116172982112072278287, −9.766920152608852927796353253645, −8.58116470303610540130445721360, −7.97138486413638500204778328873, −7.18984020818666906799481014899, −6.47885276000459571709592568399, −5.43974557618345361593007107328, −4.39405127798472042303891526617, −3.67680776242757788854633768288, −3.120754798922729742616082364145, −2.40785132808461401827423896149, −1.03825733659848533780468331498,
1.03825733659848533780468331498, 2.40785132808461401827423896149, 3.120754798922729742616082364145, 3.67680776242757788854633768288, 4.39405127798472042303891526617, 5.43974557618345361593007107328, 6.47885276000459571709592568399, 7.18984020818666906799481014899, 7.97138486413638500204778328873, 8.58116470303610540130445721360, 9.766920152608852927796353253645, 10.55797026116172982112072278287, 11.283886108127963377857183441934, 12.37268798493004718145995272131, 12.95541582369924354887084472964, 13.34330175417570285960751075531, 14.31830449861400584737582296853, 15.195521258546149673767869586061, 15.46565119145758220462093574284, 16.23447296547740273319079466184, 16.80753324942461502380856885938, 18.60501381875913101633718839974, 18.87329144144293239585017466762, 19.64124649610267716416998771591, 20.27405781989555772171566192103
