| L(s) = 1 | + 2-s − 3-s + 4-s + 1.49·5-s − 6-s + 8-s + 9-s + 1.49·10-s + 0.794·11-s − 12-s − 2·13-s − 1.49·15-s + 16-s − 4.49·17-s + 18-s − 19-s + 1.49·20-s + 0.794·22-s + 3.15·23-s − 24-s − 2.77·25-s − 2·26-s − 27-s − 6.23·29-s − 1.49·30-s − 9.01·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.666·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.471·10-s + 0.239·11-s − 0.288·12-s − 0.554·13-s − 0.385·15-s + 0.250·16-s − 1.08·17-s + 0.235·18-s − 0.229·19-s + 0.333·20-s + 0.169·22-s + 0.657·23-s − 0.204·24-s − 0.555·25-s − 0.392·26-s − 0.192·27-s − 1.15·29-s − 0.272·30-s − 1.61·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.49T + 5T^{2} \) |
| 11 | \( 1 - 0.794T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 5.42T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−7.52253779491042261896289071133, −6.83576933335418099119606801726, −6.33151896415599929983933744379, −5.48739121668976123968325811282, −5.05263062628276835419735551636, −4.18649017871481545622757212343, −3.39941091907682520452109981553, −2.24719696088477330660576997310, −1.63431299418823313701778749469, 0,
1.63431299418823313701778749469, 2.24719696088477330660576997310, 3.39941091907682520452109981553, 4.18649017871481545622757212343, 5.05263062628276835419735551636, 5.48739121668976123968325811282, 6.33151896415599929983933744379, 6.83576933335418099119606801726, 7.52253779491042261896289071133
