| L(s) = 1 | + 1.47·2-s − 0.631·3-s + 0.173·4-s + 5-s − 0.930·6-s + 7-s − 2.69·8-s − 2.60·9-s + 1.47·10-s − 2.63·11-s − 0.109·12-s + 3.76·13-s + 1.47·14-s − 0.631·15-s − 4.31·16-s − 6.95·17-s − 3.83·18-s + 5.14·19-s + 0.173·20-s − 0.631·21-s − 3.88·22-s + 7.04·23-s + 1.70·24-s + 25-s + 5.55·26-s + 3.53·27-s + 0.173·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.364·3-s + 0.0867·4-s + 0.447·5-s − 0.380·6-s + 0.377·7-s − 0.952·8-s − 0.867·9-s + 0.466·10-s − 0.795·11-s − 0.0316·12-s + 1.04·13-s + 0.394·14-s − 0.163·15-s − 1.07·16-s − 1.68·17-s − 0.903·18-s + 1.18·19-s + 0.0387·20-s − 0.137·21-s − 0.829·22-s + 1.46·23-s + 0.347·24-s + 0.200·25-s + 1.08·26-s + 0.680·27-s + 0.0327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 3 | \( 1 + 0.631T + 3T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 - 5.14T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 + 0.488T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.99T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 9.51T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−7.17529616226622946737761656058, −6.60757600486148466661664714800, −5.80126969281544166195214950325, −5.38117491102113740406518470779, −4.81392143087139182887004222303, −4.08600258518521017327524116602, −2.99943985661483385040502496372, −2.68698355231459862928917332012, −1.31138484072594552358892309575, 0,
1.31138484072594552358892309575, 2.68698355231459862928917332012, 2.99943985661483385040502496372, 4.08600258518521017327524116602, 4.81392143087139182887004222303, 5.38117491102113740406518470779, 5.80126969281544166195214950325, 6.60757600486148466661664714800, 7.17529616226622946737761656058
