| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 13-s − 17-s − 21-s + 23-s − 27-s − 29-s + 31-s + 33-s + 37-s − 39-s − 41-s + 43-s + 47-s + 49-s + 51-s + 53-s + 59-s + 61-s + 63-s − 67-s − 69-s + 71-s − 73-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 13-s − 17-s − 21-s + 23-s − 27-s − 29-s + 31-s + 33-s + 37-s − 39-s − 41-s + 43-s + 47-s + 49-s + 51-s + 53-s + 59-s + 61-s + 63-s − 67-s − 69-s + 71-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.026363169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026363169\) |
| \(L(1)\) |
\(\approx\) |
\(0.8939440530\) |
| \(L(1)\) |
\(\approx\) |
\(0.8939440530\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 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
−24.26803349235082566902014341649, −23.68744571912713879965382467033, −22.927624764457365140888182097444, −21.96584427320489311321974889959, −21.04189927040236197438724580006, −20.54025372934600249548169557650, −18.98604967176938932496829954784, −18.215699705731517176112656661257, −17.61594159804548784158803326780, −16.69136411273428757781121040896, −15.686670909534621716029584998072, −15.04125605012948133885270202781, −13.57469839645539401504365027176, −12.954783842613575420119410424243, −11.69761462007320007733512830437, −11.03592307182931153725036293892, −10.39321143800167989689554989531, −8.98446163993967406087226796894, −7.93183531466716919920684097187, −6.92532686274331445064505463953, −5.79445829465544870140371708503, −4.98039999099752026737427355448, −4.03794531127032443628703529221, −2.31710381040277474888182916785, −0.99769864034816801341756551625,
0.99769864034816801341756551625, 2.31710381040277474888182916785, 4.03794531127032443628703529221, 4.98039999099752026737427355448, 5.79445829465544870140371708503, 6.92532686274331445064505463953, 7.93183531466716919920684097187, 8.98446163993967406087226796894, 10.39321143800167989689554989531, 11.03592307182931153725036293892, 11.69761462007320007733512830437, 12.954783842613575420119410424243, 13.57469839645539401504365027176, 15.04125605012948133885270202781, 15.686670909534621716029584998072, 16.69136411273428757781121040896, 17.61594159804548784158803326780, 18.215699705731517176112656661257, 18.98604967176938932496829954784, 20.54025372934600249548169557650, 21.04189927040236197438724580006, 21.96584427320489311321974889959, 22.927624764457365140888182097444, 23.68744571912713879965382467033, 24.26803349235082566902014341649
