| L(s) = 1 | − 3-s + 9-s − 11-s − 13-s + 17-s + 19-s − 23-s − 27-s − 29-s − 31-s + 33-s + 37-s + 39-s − 41-s + 43-s + 47-s − 51-s + 53-s − 57-s + 59-s + 61-s + 67-s + 69-s + 71-s + 73-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | − 3-s + 9-s − 11-s − 13-s + 17-s + 19-s − 23-s − 27-s − 29-s − 31-s + 33-s + 37-s + 39-s − 41-s + 43-s + 47-s − 51-s + 53-s − 57-s + 59-s + 61-s + 67-s + 69-s + 71-s + 73-s + 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036313191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036313191\) |
| \(L(1)\) |
\(\approx\) |
\(0.7509842836\) |
| \(L(1)\) |
\(\approx\) |
\(0.7509842836\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 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
−25.3483861660241312048265784897, −24.14810134012836111875180193870, −23.72123309368622894187468053763, −22.600110359053398240858954658407, −21.954615974316446452411210053452, −21.00195801443738289667496315346, −19.988673135709712473922961788276, −18.66674470566623331820341404342, −18.15238404677770442203636246105, −17.04597309992957598752004950112, −16.317546172077968592974945024302, −15.41483877585247123765538331835, −14.26395274226807174975302474976, −13.023084968141439516557514435514, −12.23257421851890366442624531660, −11.345351386819402741143704520265, −10.25107861522765682790130608499, −9.58541897791053040555980484876, −7.8444584284517010096034187276, −7.164891206540729639032036712232, −5.69400864325719275211064928449, −5.16940754523429893265909483831, −3.79172986982335633751636325007, −2.22607177487909463762206315362, −0.64271518477675268997729318603,
0.64271518477675268997729318603, 2.22607177487909463762206315362, 3.79172986982335633751636325007, 5.16940754523429893265909483831, 5.69400864325719275211064928449, 7.164891206540729639032036712232, 7.8444584284517010096034187276, 9.58541897791053040555980484876, 10.25107861522765682790130608499, 11.345351386819402741143704520265, 12.23257421851890366442624531660, 13.023084968141439516557514435514, 14.26395274226807174975302474976, 15.41483877585247123765538331835, 16.317546172077968592974945024302, 17.04597309992957598752004950112, 18.15238404677770442203636246105, 18.66674470566623331820341404342, 19.988673135709712473922961788276, 21.00195801443738289667496315346, 21.954615974316446452411210053452, 22.600110359053398240858954658407, 23.72123309368622894187468053763, 24.14810134012836111875180193870, 25.3483861660241312048265784897
