| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2.34·7-s + 8-s + 9-s + 10-s − 2.12·11-s + 12-s − 5.29·13-s − 2.34·14-s + 15-s + 16-s + 18-s + 8.51·19-s + 20-s − 2.34·21-s − 2.12·22-s − 2.83·23-s + 24-s + 25-s − 5.29·26-s + 27-s − 2.34·28-s − 2.69·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.887·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.639·11-s + 0.288·12-s − 1.46·13-s − 0.627·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 1.95·19-s + 0.223·20-s − 0.512·21-s − 0.452·22-s − 0.591·23-s + 0.204·24-s + 0.200·25-s − 1.03·26-s + 0.192·27-s − 0.443·28-s − 0.500·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 19 | \( 1 - 8.51T + 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 + 2.69T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 6.06T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.573T + 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 + 0.0837T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.50262508111739543282296915641, −6.73035732401241461956626364467, −6.00971124922082683876903359119, −5.13293599522968322518527895896, −4.87234883242456991378946785112, −3.58428389503713129395055574156, −3.16702207742172259158641565232, −2.44011283825417396580441599937, −1.59015653753308916951349240472, 0,
1.59015653753308916951349240472, 2.44011283825417396580441599937, 3.16702207742172259158641565232, 3.58428389503713129395055574156, 4.87234883242456991378946785112, 5.13293599522968322518527895896, 6.00971124922082683876903359119, 6.73035732401241461956626364467, 7.50262508111739543282296915641
