| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s + 13-s − 15-s − 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s − 41-s − 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s + 13-s − 15-s − 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s − 41-s − 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.290573151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290573151\) |
| \(L(1)\) |
\(\approx\) |
\(1.167563714\) |
| \(L(1)\) |
\(\approx\) |
\(1.167563714\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 181 | \( 1 \) |
| good | 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
−22.29575800104743640162798338546, −21.448919977572433393894916609745, −21.02002629992879818351926627608, −20.193375917981784217369164340150, −18.53928794195841621260928168130, −18.252800687733409238791099417292, −17.535075021264543336283176991380, −16.85439170598201781760887400024, −15.810114394172957069635385336997, −15.18385637279990271600427015787, −13.76452849188401551411702526229, −13.42725143391714324300542028928, −12.37075291789788306566047689464, −11.31311534736387836740181955542, −10.78845070563402830304788906382, −10.004505792403861556258187819472, −8.91340604949901560690482899703, −7.91220416933984546038311828143, −6.795852288378813778989562995211, −5.99250090566759657062705145828, −5.10122896579355186681116793409, −4.57844561296957854677825528961, −2.909352839269336899866138441057, −1.69769503889243059387086141338, −0.8523146241373806874300665396,
0.8523146241373806874300665396, 1.69769503889243059387086141338, 2.909352839269336899866138441057, 4.57844561296957854677825528961, 5.10122896579355186681116793409, 5.99250090566759657062705145828, 6.795852288378813778989562995211, 7.91220416933984546038311828143, 8.91340604949901560690482899703, 10.004505792403861556258187819472, 10.78845070563402830304788906382, 11.31311534736387836740181955542, 12.37075291789788306566047689464, 13.42725143391714324300542028928, 13.76452849188401551411702526229, 15.18385637279990271600427015787, 15.810114394172957069635385336997, 16.85439170598201781760887400024, 17.535075021264543336283176991380, 18.252800687733409238791099417292, 18.53928794195841621260928168130, 20.193375917981784217369164340150, 21.02002629992879818351926627608, 21.448919977572433393894916609745, 22.29575800104743640162798338546
