| L(s) = 1 | + 1.63·2-s + 3-s + 0.667·4-s − 0.873·5-s + 1.63·6-s + 4.38·7-s − 2.17·8-s + 9-s − 1.42·10-s − 3.80·11-s + 0.667·12-s − 1.80·13-s + 7.15·14-s − 0.873·15-s − 4.88·16-s + 4.43·17-s + 1.63·18-s − 0.0799·19-s − 0.583·20-s + 4.38·21-s − 6.22·22-s − 5.01·23-s − 2.17·24-s − 4.23·25-s − 2.95·26-s + 27-s + 2.92·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.577·3-s + 0.333·4-s − 0.390·5-s + 0.666·6-s + 1.65·7-s − 0.769·8-s + 0.333·9-s − 0.451·10-s − 1.14·11-s + 0.192·12-s − 0.501·13-s + 1.91·14-s − 0.225·15-s − 1.22·16-s + 1.07·17-s + 0.384·18-s − 0.0183·19-s − 0.130·20-s + 0.956·21-s − 1.32·22-s − 1.04·23-s − 0.444·24-s − 0.847·25-s − 0.579·26-s + 0.192·27-s + 0.552·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227626293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227626293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 5 | \( 1 + 0.873T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 + 0.0799T + 19T^{2} \) |
| 23 | \( 1 + 5.01T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 + 0.0919T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 + 0.583T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 0.0799T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 + 6.02T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−12.50218780370839342245958826587, −11.80451223278812892932890331070, −10.75993207137455241614592044068, −9.461995277019954181995264320996, −8.047747998662067904628267630100, −7.67607952565961795904728471905, −5.71533954470351759201949442004, −4.84961506982248912812360369223, −3.83831486766162639447400563118, −2.33556550242256407561875745858,
2.33556550242256407561875745858, 3.83831486766162639447400563118, 4.84961506982248912812360369223, 5.71533954470351759201949442004, 7.67607952565961795904728471905, 8.047747998662067904628267630100, 9.461995277019954181995264320996, 10.75993207137455241614592044068, 11.80451223278812892932890331070, 12.50218780370839342245958826587
