| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 22-s − 23-s + 26-s − 28-s + 29-s − 31-s + 32-s + 34-s − 37-s − 38-s + 41-s + 43-s + 44-s − 46-s + 49-s + 52-s + 53-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 22-s − 23-s + 26-s − 28-s + 29-s − 31-s + 32-s + 34-s − 37-s − 38-s + 41-s + 43-s + 44-s − 46-s + 49-s + 52-s + 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.808862036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808862036\) |
| \(L(1)\) |
\(\approx\) |
\(1.968924732\) |
| \(L(1)\) |
\(\approx\) |
\(1.968924732\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 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 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.72980314573962794097944512167, −21.879755118295648008583331421498, −21.19452216928255823486288978135, −20.27481736707147400341783092848, −19.494756190017808532502634547112, −18.89311477217641530632568562835, −17.56952098490414915656309953196, −16.50584260927236057961656284491, −16.10647600025378402702981244840, −15.13779248215141000708305575552, −14.226549722172903961488103802133, −13.624664818522539746901554080085, −12.560834581782132711759683578064, −12.15751514001840131727326505112, −11.02245877562729968182074051202, −10.24117986238998913547461521431, −9.1896891050916401336162774572, −8.077672711355664912432693917959, −6.88473547442486461271672264925, −6.25314745824989517722403336489, −5.51869501169242862433346076067, −4.04440306472949600591686723114, −3.664881697504305127270110103807, −2.48939541126726958458081669450, −1.24400706606971551933883626209,
1.24400706606971551933883626209, 2.48939541126726958458081669450, 3.664881697504305127270110103807, 4.04440306472949600591686723114, 5.51869501169242862433346076067, 6.25314745824989517722403336489, 6.88473547442486461271672264925, 8.077672711355664912432693917959, 9.1896891050916401336162774572, 10.24117986238998913547461521431, 11.02245877562729968182074051202, 12.15751514001840131727326505112, 12.560834581782132711759683578064, 13.624664818522539746901554080085, 14.226549722172903961488103802133, 15.13779248215141000708305575552, 16.10647600025378402702981244840, 16.50584260927236057961656284491, 17.56952098490414915656309953196, 18.89311477217641530632568562835, 19.494756190017808532502634547112, 20.27481736707147400341783092848, 21.19452216928255823486288978135, 21.879755118295648008583331421498, 22.72980314573962794097944512167
