| L(s) = 1 | − 0.638·2-s + 3·3-s − 7.59·4-s − 13.5·5-s − 1.91·6-s + 17.9·7-s + 9.95·8-s + 9·9-s + 8.65·10-s − 26.8·11-s − 22.7·12-s − 0.330·13-s − 11.4·14-s − 40.6·15-s + 54.3·16-s + 55.0·17-s − 5.74·18-s + 124.·19-s + 102.·20-s + 53.8·21-s + 17.1·22-s + 106.·23-s + 29.8·24-s + 58.5·25-s + 0.211·26-s + 27·27-s − 136.·28-s + ⋯ |
| L(s) = 1 | − 0.225·2-s + 0.577·3-s − 0.948·4-s − 1.21·5-s − 0.130·6-s + 0.968·7-s + 0.440·8-s + 0.333·9-s + 0.273·10-s − 0.735·11-s − 0.547·12-s − 0.00705·13-s − 0.218·14-s − 0.699·15-s + 0.849·16-s + 0.785·17-s − 0.0752·18-s + 1.50·19-s + 1.14·20-s + 0.559·21-s + 0.166·22-s + 0.961·23-s + 0.254·24-s + 0.468·25-s + 0.00159·26-s + 0.192·27-s − 0.919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.340919647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340919647\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 + 0.638T + 8T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 7 | \( 1 - 17.9T + 343T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.330T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 37.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 49.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 48.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 21.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 435.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 384.T + 2.26e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 756.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 11.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 623.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−11.98864635110878574220055068869, −11.03342241189356000989284762305, −9.884354413205704895248906754276, −8.845684279583697454481698814735, −7.81970507795242264468685259834, −7.61837198528647137327605461980, −5.29350006617258335716887258194, −4.37571347825475939988118154905, −3.17247557434157838661714019658, −0.944057616828019379961533062061,
0.944057616828019379961533062061, 3.17247557434157838661714019658, 4.37571347825475939988118154905, 5.29350006617258335716887258194, 7.61837198528647137327605461980, 7.81970507795242264468685259834, 8.845684279583697454481698814735, 9.884354413205704895248906754276, 11.03342241189356000989284762305, 11.98864635110878574220055068869
