| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.241278243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241278243\) |
| \(L(1)\) |
\(\approx\) |
\(1.384281850\) |
| \(L(1)\) |
\(\approx\) |
\(1.384281850\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1549 | \( 1 \) |
| 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
−20.50484700739128629320374765641, −19.71992352852300465318656488629, −19.02211810995428327913420562958, −18.437816799897592178874206183711, −17.4067993177331441373589600901, −17.14568769590451241086190117655, −16.28808927645705975688028839216, −14.956765163013897699125422335047, −14.67310246498837184018956451283, −14.140999983839889249153064361167, −12.881474129137217605616390690723, −12.254811051487039879652888724861, −11.11189047898003555960711614292, −10.44094184018796783222510700398, −9.48204064761694816756793176195, −9.17582680890852224949842447909, −8.34279133704592229728604740091, −7.47622334387101464667608337529, −6.89907966582622831407518566337, −5.808744318723094790843670981279, −4.81847381642394332313317000508, −3.60340328222588555517318078664, −2.553901708283328176231790540187, −1.83139746511306555916924115218, −1.23195550353945233118489579978,
1.23195550353945233118489579978, 1.83139746511306555916924115218, 2.553901708283328176231790540187, 3.60340328222588555517318078664, 4.81847381642394332313317000508, 5.808744318723094790843670981279, 6.89907966582622831407518566337, 7.47622334387101464667608337529, 8.34279133704592229728604740091, 9.17582680890852224949842447909, 9.48204064761694816756793176195, 10.44094184018796783222510700398, 11.11189047898003555960711614292, 12.254811051487039879652888724861, 12.881474129137217605616390690723, 14.140999983839889249153064361167, 14.67310246498837184018956451283, 14.956765163013897699125422335047, 16.28808927645705975688028839216, 17.14568769590451241086190117655, 17.4067993177331441373589600901, 18.437816799897592178874206183711, 19.02211810995428327913420562958, 19.71992352852300465318656488629, 20.50484700739128629320374765641
