| L(s) = 1 | + 2.39·2-s + 1.73·4-s − 9.98·5-s + 10.2·7-s − 5.41·8-s + 9·9-s − 23.9·10-s + 24.5·14-s − 19.9·16-s + 21.5·18-s − 11.0·19-s − 17.3·20-s + 74.6·25-s + 17.8·28-s − 31·31-s − 26.0·32-s − 102.·35-s + 15.6·36-s − 26.4·38-s + 54.0·40-s + 12.3·41-s − 89.8·45-s − 30·47-s + 56.0·49-s + 178.·50-s − 55.5·56-s + 108.·59-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.434·4-s − 1.99·5-s + 1.46·7-s − 0.677·8-s + 9-s − 2.39·10-s + 1.75·14-s − 1.24·16-s + 1.19·18-s − 0.581·19-s − 0.867·20-s + 2.98·25-s + 0.636·28-s − 31-s − 0.814·32-s − 2.92·35-s + 0.434·36-s − 0.696·38-s + 1.35·40-s + 0.302·41-s − 1.99·45-s − 0.638·47-s + 1.14·49-s + 3.57·50-s − 0.991·56-s + 1.83·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.394505437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394505437\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + 31T \) |
| good | 2 | \( 1 - 2.39T + 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 + 9.98T + 25T^{2} \) |
| 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 11.0T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 12.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 30T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 108.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 10T + 4.48e3T^{2} \) |
| 71 | \( 1 + 112.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 104.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.15999683787479068134464918491, −15.14012756956488618529647969673, −14.57332882175870503705594733286, −12.89679906733694773002436370268, −11.91154700257486403052897418123, −11.03650665469179773160258904066, −8.469553737557101053224507913069, −7.26143369203609402753904570035, −4.80855861602909213501152722686, −3.92924440423807003687096439452,
3.92924440423807003687096439452, 4.80855861602909213501152722686, 7.26143369203609402753904570035, 8.469553737557101053224507913069, 11.03650665469179773160258904066, 11.91154700257486403052897418123, 12.89679906733694773002436370268, 14.57332882175870503705594733286, 15.14012756956488618529647969673, 16.15999683787479068134464918491
