L(s) = 1 | + 1.80·2-s + 3-s + 1.24·4-s − 5-s + 1.80·6-s − 1.41·7-s − 1.36·8-s + 9-s − 1.80·10-s − 0.461·11-s + 1.24·12-s − 0.961·13-s − 2.54·14-s − 15-s − 4.94·16-s + 6.22·17-s + 1.80·18-s + 0.763·19-s − 1.24·20-s − 1.41·21-s − 0.830·22-s − 4.92·23-s − 1.36·24-s + 25-s − 1.73·26-s + 27-s − 1.75·28-s + ⋯ |
L(s) = 1 | + 1.27·2-s + 0.577·3-s + 0.620·4-s − 0.447·5-s + 0.734·6-s − 0.534·7-s − 0.483·8-s + 0.333·9-s − 0.569·10-s − 0.139·11-s + 0.358·12-s − 0.266·13-s − 0.680·14-s − 0.258·15-s − 1.23·16-s + 1.50·17-s + 0.424·18-s + 0.175·19-s − 0.277·20-s − 0.308·21-s − 0.177·22-s − 1.02·23-s − 0.278·24-s + 0.200·25-s − 0.339·26-s + 0.192·27-s − 0.331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.461T + 11T^{2} \) |
| 13 | \( 1 + 0.961T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 8.68T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 7.54T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 1.99T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 0.239T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−7.84679809393954943367655069188, −6.70329640828921152280105948514, −6.37140791111449629734442624440, −5.27030949893263871346045863497, −4.88741109346350386571940539505, −3.87858669279364415023668064207, −3.34299325775300546115889673107, −2.85056262701737049632578084209, −1.62769601864727562442905788783, 0,
1.62769601864727562442905788783, 2.85056262701737049632578084209, 3.34299325775300546115889673107, 3.87858669279364415023668064207, 4.88741109346350386571940539505, 5.27030949893263871346045863497, 6.37140791111449629734442624440, 6.70329640828921152280105948514, 7.84679809393954943367655069188