L(s) = 1 | + i·3-s + i·5-s − 9-s + i·11-s − i·13-s − 15-s − 17-s + i·19-s + 23-s − 25-s − i·27-s + i·29-s + 31-s − 33-s − i·37-s + ⋯ |
L(s) = 1 | + i·3-s + i·5-s − 9-s + i·11-s − i·13-s − 15-s − 17-s + i·19-s + 23-s − 25-s − i·27-s + i·29-s + 31-s − 33-s − i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5190104816 + 0.7767540777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5190104816 + 0.7767540777i\) |
\(L(1)\) |
\(\approx\) |
\(0.8245509788 + 0.5193117243i\) |
\(L(1)\) |
\(\approx\) |
\(0.8245509788 + 0.5193117243i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 \) |
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
−28.85193244516978836270454980920, −28.57541780434474264938900745983, −27.02553478297200288518874312825, −25.924699456397585631989049519770, −24.60103785243527176922300546741, −24.24961548457097367846675527583, −23.26316655023189557062140061974, −21.84776279823434634069388578753, −20.728161141329723245642550526, −19.57430292080512069177088345570, −18.882488516071708181966557704053, −17.49037855556254891523836285025, −16.76148568216737025438627203606, −15.48547491955593630335081798038, −13.77084627165018342045410597304, −13.29996727930137831268798548768, −12.00915276340071234452751618091, −11.159719522788952829478102150137, −9.12639614454421620966515541204, −8.45343516246558317363233138980, −7.02865518583676216037116968366, −5.870145040985107174055441987472, −4.4519511828607911217972207146, −2.52432013634000416907028744949, −0.95440583083808087788258509901,
2.52614859787615639196149570702, 3.74982197900302381674846027535, 5.09857276679580071930394334453, 6.492402029192535482367767543771, 7.88235690801071974420220583016, 9.38126204601217236206889106590, 10.39643145950399933576607395080, 11.14440194703323902540720237885, 12.638028308436391440051830364420, 14.21912234372911825947542137116, 15.0544412628717394331283854054, 15.80929098584573761751079245558, 17.279889273093895409049890626009, 18.09824903961264985724214285945, 19.547049900248133264553362700858, 20.51162864233845094970984008578, 21.56663056469619869141208186971, 22.70952661139529751412062824293, 22.9928545408680475656701106032, 24.93680669808358249152344519466, 25.81219431338083694260366864747, 26.76664990166874405741830016210, 27.50313887018022612434013485386, 28.58474140979756436103894575327, 29.72558423155565883818737348742