| L(s) = 1 | + 0.615·2-s + 2.77·3-s − 1.62·4-s − 5-s + 1.70·6-s − 7-s − 2.22·8-s + 4.67·9-s − 0.615·10-s + 0.916·11-s − 4.49·12-s − 0.361·13-s − 0.615·14-s − 2.77·15-s + 1.86·16-s + 1.55·17-s + 2.88·18-s − 3.76·19-s + 1.62·20-s − 2.77·21-s + 0.564·22-s − 5.59·23-s − 6.17·24-s + 25-s − 0.222·26-s + 4.65·27-s + 1.62·28-s + ⋯ |
| L(s) = 1 | + 0.435·2-s + 1.59·3-s − 0.810·4-s − 0.447·5-s + 0.696·6-s − 0.377·7-s − 0.788·8-s + 1.55·9-s − 0.194·10-s + 0.276·11-s − 1.29·12-s − 0.100·13-s − 0.164·14-s − 0.715·15-s + 0.467·16-s + 0.377·17-s + 0.678·18-s − 0.863·19-s + 0.362·20-s − 0.604·21-s + 0.120·22-s − 1.16·23-s − 1.26·24-s + 0.200·25-s − 0.0436·26-s + 0.895·27-s + 0.306·28-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 | \( 1 - 0.615T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 11 | \( 1 - 0.916T + 11T^{2} \) |
| 13 | \( 1 + 0.361T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 5.59T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 + 0.0665T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 - 8.25T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 + 0.926T + 83T^{2} \) |
| 89 | \( 1 + 0.0548T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.79288291895997404017003770819, −6.81720951894143655077315600796, −6.20108252877502531582790875739, −5.15457099581127228746217472981, −4.33909132504271621825811103279, −3.87930582804817685626996077300, −3.20578427582305161528395259897, −2.57794194787909923672570713482, −1.45962776914882734449627338863, 0,
1.45962776914882734449627338863, 2.57794194787909923672570713482, 3.20578427582305161528395259897, 3.87930582804817685626996077300, 4.33909132504271621825811103279, 5.15457099581127228746217472981, 6.20108252877502531582790875739, 6.81720951894143655077315600796, 7.79288291895997404017003770819
