| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.579721996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.579721996\) |
| \(L(1)\) |
\(\approx\) |
\(2.955129663\) |
| \(L(1)\) |
\(\approx\) |
\(2.955129663\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 191 | \( 1 \) |
| 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
−26.20671702581754744588596270571, −25.672925909801622283463701368019, −25.21387968517665194462959444351, −23.96739789306231627356779495973, −23.05601835072699892888746793897, −21.937688453979384624516750396415, −20.9581901343345070713664576529, −20.668484608510148778508733792870, −19.25810236745835591265442679451, −18.52347890531904726423229021195, −16.81222551987540191658500819391, −15.891395788878434294164149714965, −14.94100760942671095841657288341, −13.96602844677867602811394234382, −13.01440405989863879105767646718, −12.81078478930838625255705263221, −10.788245772618255701473717594704, −9.972269360339284082981310574375, −8.75920985641220149799706079523, −7.35710426349626794442192518834, −6.28638578137027435179156948798, −5.23030598939777040470115698044, −3.64787749359291744095988493307, −2.83268729135811392483472343303, −1.656708874838992162047069700557,
1.656708874838992162047069700557, 2.83268729135811392483472343303, 3.64787749359291744095988493307, 5.23030598939777040470115698044, 6.28638578137027435179156948798, 7.35710426349626794442192518834, 8.75920985641220149799706079523, 9.972269360339284082981310574375, 10.788245772618255701473717594704, 12.81078478930838625255705263221, 13.01440405989863879105767646718, 13.96602844677867602811394234382, 14.94100760942671095841657288341, 15.891395788878434294164149714965, 16.81222551987540191658500819391, 18.52347890531904726423229021195, 19.25810236745835591265442679451, 20.668484608510148778508733792870, 20.9581901343345070713664576529, 21.937688453979384624516750396415, 23.05601835072699892888746793897, 23.96739789306231627356779495973, 25.21387968517665194462959444351, 25.672925909801622283463701368019, 26.20671702581754744588596270571
