| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6041301740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6041301740\) |
| \(L(1)\) |
\(\approx\) |
\(0.5822647269\) |
| \(L(1)\) |
\(\approx\) |
\(0.5822647269\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 131 | \( 1 \) |
| 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
−21.643854680552555301192714126748, −20.44272391113410182471020631254, −19.92169189064306442790411838805, −18.83010742442415810256040449186, −18.003766108094701339437217595652, −17.69982259162103491864172617622, −16.52385007532382300227674855757, −15.9417831579413604759971176232, −15.23556629755502099353464700712, −14.40523744501904383082410149788, −13.26487900513323029739349062001, −12.36775905311646326302685909805, −11.743603227724334456571920879257, −11.092571576537991762900592409111, −10.41823238426689634838168813100, −9.38349125143020031787156121547, −8.02816625081039531667418618903, −7.654750391010477843428040969000, −6.743119212196838670308481963510, −5.54775756874598394947149122848, −4.771047254068149159905184517459, −4.28289366901505056249533157725, −2.86024669535402306159917109059, −1.65080018415708969708352242704, −0.37808103902154541988554540092,
0.37808103902154541988554540092, 1.65080018415708969708352242704, 2.86024669535402306159917109059, 4.28289366901505056249533157725, 4.771047254068149159905184517459, 5.54775756874598394947149122848, 6.743119212196838670308481963510, 7.654750391010477843428040969000, 8.02816625081039531667418618903, 9.38349125143020031787156121547, 10.41823238426689634838168813100, 11.092571576537991762900592409111, 11.743603227724334456571920879257, 12.36775905311646326302685909805, 13.26487900513323029739349062001, 14.40523744501904383082410149788, 15.23556629755502099353464700712, 15.9417831579413604759971176232, 16.52385007532382300227674855757, 17.69982259162103491864172617622, 18.003766108094701339437217595652, 18.83010742442415810256040449186, 19.92169189064306442790411838805, 20.44272391113410182471020631254, 21.643854680552555301192714126748
