L(s) = 1 | + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s − 21-s + 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.809 − 0.587i)31-s + (0.809 + 0.587i)35-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s − 21-s + 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.809 − 0.587i)31-s + (0.809 + 0.587i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8873123181 + 0.4779865549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8873123181 + 0.4779865549i\) |
\(L(1)\) |
\(\approx\) |
\(1.001420540 + 0.3227405149i\) |
\(L(1)\) |
\(\approx\) |
\(1.001420540 + 0.3227405149i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−30.25202639817529024507248100887, −29.52165659944778518926318249598, −28.64016833973288808684928223721, −27.243159204910453363453460425358, −25.98918555782680332776654256532, −25.169824596028928880488958956769, −24.004203274344361597612995481788, −23.0652868225934858926999143724, −22.11150354881355887057883427140, −20.68383291008391938717770623390, −19.60660805317336798803179186391, −18.222087230868698300362340962206, −17.703881316413473183376618770281, −16.60119965734599724555729547701, −14.833893542102604704918239361, −13.58339224555293266776845042573, −13.1123906325926113623702561752, −11.27250328462869604211085300400, −10.59233933498749200573158606595, −8.83869567507338897651220114347, −7.29144661096309426908815016015, −6.513973490625909545861746738274, −5.07885796113543017750302778475, −2.95769092282730251397806884138, −1.36391045074167694049536803234,
2.0402878516846803363646896500, 3.99746052763048929477050903683, 5.341810199486156004868378140194, 6.194815391562907522742974636726, 8.596617195202677711721682097473, 9.24729213184655978584207487491, 10.58675809399275808096180260214, 11.74516962601399400143715055862, 13.059426931199010889492380000220, 14.47927988695861736536003940929, 15.53938915139503183164600164581, 16.63595796036717431838305712184, 17.553312413568745776677832899461, 18.76545533252955094572267717068, 20.46162621346778344028661022340, 21.25060212890845185863158748416, 21.91927595150137256852535890384, 23.19569902821563278588781011388, 24.53724835795915504411979224066, 25.50802110289202316829722305691, 26.55974144906941114936055059899, 27.88456088410504525301877431852, 28.4412015651450240094808394495, 29.35106018367485817750632756029, 31.02652445292420804418057384800