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 | \( 1 \) |
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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
−29.72558423155565883818737348742, −28.58474140979756436103894575327, −27.50313887018022612434013485386, −26.76664990166874405741830016210, −25.81219431338083694260366864747, −24.93680669808358249152344519466, −22.9928545408680475656701106032, −22.70952661139529751412062824293, −21.56663056469619869141208186971, −20.51162864233845094970984008578, −19.547049900248133264553362700858, −18.09824903961264985724214285945, −17.279889273093895409049890626009, −15.80929098584573761751079245558, −15.0544412628717394331283854054, −14.21912234372911825947542137116, −12.638028308436391440051830364420, −11.14440194703323902540720237885, −10.39643145950399933576607395080, −9.38126204601217236206889106590, −7.88235690801071974420220583016, −6.492402029192535482367767543771, −5.09857276679580071930394334453, −3.74982197900302381674846027535, −2.52614859787615639196149570702,
0.95440583083808087788258509901, 2.52432013634000416907028744949, 4.4519511828607911217972207146, 5.870145040985107174055441987472, 7.02865518583676216037116968366, 8.45343516246558317363233138980, 9.12639614454421620966515541204, 11.159719522788952829478102150137, 12.00915276340071234452751618091, 13.29996727930137831268798548768, 13.77084627165018342045410597304, 15.48547491955593630335081798038, 16.76148568216737025438627203606, 17.49037855556254891523836285025, 18.882488516071708181966557704053, 19.57430292080512069177088345570, 20.728161141329723245642550526, 21.84776279823434634069388578753, 23.26316655023189557062140061974, 24.24961548457097367846675527583, 24.60103785243527176922300546741, 25.924699456397585631989049519770, 27.02553478297200288518874312825, 28.57541780434474264938900745983, 28.85193244516978836270454980920