| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.631358902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631358902\) |
| \(L(1)\) |
\(\approx\) |
\(1.130837073\) |
| \(L(1)\) |
\(\approx\) |
\(1.130837073\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1069 | \( 1 \) |
| 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
−21.335460833047656495140365130206, −20.409322519220527118964950520557, −20.06378608959351325520470004059, −19.008272429606601034939503437, −18.24394416848617688462810098572, −18.11593292037744131507207605725, −16.59240436530444804047272979750, −16.17980026536149033886025146918, −15.4407942548402582478739321225, −14.43328298143321991171241748074, −13.55548301000671350748392684138, −12.972117579010113188385790385892, −12.00982186723874336149593319022, −10.65280932372088875399381707475, −9.932688443406725796468351208004, −9.65339308149452060061459471800, −8.64211878771895071505699268366, −7.97711934290423326724046601309, −7.030901446065347669681940913361, −6.19405389479680696127967076793, −5.33758266153677494701938146712, −3.53854593842573313294799984254, −2.949263370571121883871299420286, −2.0421275606331166992099861817, −1.049324784840336105248480366706,
1.049324784840336105248480366706, 2.0421275606331166992099861817, 2.949263370571121883871299420286, 3.53854593842573313294799984254, 5.33758266153677494701938146712, 6.19405389479680696127967076793, 7.030901446065347669681940913361, 7.97711934290423326724046601309, 8.64211878771895071505699268366, 9.65339308149452060061459471800, 9.932688443406725796468351208004, 10.65280932372088875399381707475, 12.00982186723874336149593319022, 12.972117579010113188385790385892, 13.55548301000671350748392684138, 14.43328298143321991171241748074, 15.4407942548402582478739321225, 16.17980026536149033886025146918, 16.59240436530444804047272979750, 18.11593292037744131507207605725, 18.24394416848617688462810098572, 19.008272429606601034939503437, 20.06378608959351325520470004059, 20.409322519220527118964950520557, 21.335460833047656495140365130206
