L(s) = 1 | − 2-s − 1.80·3-s + 4-s − 1.50·5-s + 1.80·6-s + 4.15·7-s − 8-s + 0.242·9-s + 1.50·10-s − 3.55·11-s − 1.80·12-s − 0.0848·13-s − 4.15·14-s + 2.70·15-s + 16-s − 6.67·17-s − 0.242·18-s − 7.45·19-s − 1.50·20-s − 7.47·21-s + 3.55·22-s − 1.44·23-s + 1.80·24-s − 2.73·25-s + 0.0848·26-s + 4.96·27-s + 4.15·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s − 0.672·5-s + 0.735·6-s + 1.56·7-s − 0.353·8-s + 0.0807·9-s + 0.475·10-s − 1.07·11-s − 0.519·12-s − 0.0235·13-s − 1.10·14-s + 0.699·15-s + 0.250·16-s − 1.61·17-s − 0.0570·18-s − 1.71·19-s − 0.336·20-s − 1.63·21-s + 0.758·22-s − 0.301·23-s + 0.367·24-s − 0.547·25-s + 0.0166·26-s + 0.955·27-s + 0.784·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3986204101\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3986204101\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 0.0848T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 0.906T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 0.304T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 7.20T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.00T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 + 8.74T + 73T^{2} \) |
| 79 | \( 1 - 5.64T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 4.52T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−8.314120219218130711144448977470, −8.026355906482215821630875187561, −6.92160192674711643461050885665, −6.42063036107993781119098076842, −5.40865011616381884854796402809, −4.75776952860347336783340489388, −4.16708041057448955694690820603, −2.60335223963183517296716902185, −1.82218004892836926526381236234, −0.40948767978964373260282195901,
0.40948767978964373260282195901, 1.82218004892836926526381236234, 2.60335223963183517296716902185, 4.16708041057448955694690820603, 4.75776952860347336783340489388, 5.40865011616381884854796402809, 6.42063036107993781119098076842, 6.92160192674711643461050885665, 8.026355906482215821630875187561, 8.314120219218130711144448977470