| L(s) = 1 | − 2.29·2-s − 2.84·3-s + 3.25·4-s − 5-s + 6.52·6-s − 7-s − 2.87·8-s + 5.11·9-s + 2.29·10-s + 4.04·11-s − 9.26·12-s − 5.06·13-s + 2.29·14-s + 2.84·15-s + 0.0832·16-s − 3.31·17-s − 11.7·18-s + 0.725·19-s − 3.25·20-s + 2.84·21-s − 9.27·22-s − 2.50·23-s + 8.19·24-s + 25-s + 11.6·26-s − 6.01·27-s − 3.25·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 1.64·3-s + 1.62·4-s − 0.447·5-s + 2.66·6-s − 0.377·7-s − 1.01·8-s + 1.70·9-s + 0.724·10-s + 1.22·11-s − 2.67·12-s − 1.40·13-s + 0.612·14-s + 0.735·15-s + 0.0208·16-s − 0.802·17-s − 2.76·18-s + 0.166·19-s − 0.727·20-s + 0.621·21-s − 1.97·22-s − 0.523·23-s + 1.67·24-s + 0.200·25-s + 2.27·26-s − 1.15·27-s − 0.615·28-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 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 - 0.725T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 - 0.877T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 4.81T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 1.88T + 97T^{2} \) |
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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
−7.41930126939393722263092198008, −6.81424546226548023169193776117, −6.46715646677840713342973054155, −5.67286432194811187888487717584, −4.63723032002299318766690577975, −4.19795605477639514198785974100, −2.77398920621566735628504668289, −1.70315215916881707929206393566, −0.76174899324171875454699766729, 0,
0.76174899324171875454699766729, 1.70315215916881707929206393566, 2.77398920621566735628504668289, 4.19795605477639514198785974100, 4.63723032002299318766690577975, 5.67286432194811187888487717584, 6.46715646677840713342973054155, 6.81424546226548023169193776117, 7.41930126939393722263092198008
