| L(s) = 1 | + 2-s + 4-s + 0.732·5-s − 3.73·7-s + 8-s + 0.732·10-s − 3.73·11-s − 13-s − 3.73·14-s + 16-s − 3.46·17-s + 6.46·19-s + 0.732·20-s − 3.73·22-s + 4.19·23-s − 4.46·25-s − 26-s − 3.73·28-s + 2.46·29-s − 5.46·31-s + 32-s − 3.46·34-s − 2.73·35-s − 9.46·37-s + 6.46·38-s + 0.732·40-s − 7.26·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.327·5-s − 1.41·7-s + 0.353·8-s + 0.231·10-s − 1.12·11-s − 0.277·13-s − 0.997·14-s + 0.250·16-s − 0.840·17-s + 1.48·19-s + 0.163·20-s − 0.795·22-s + 0.874·23-s − 0.892·25-s − 0.196·26-s − 0.705·28-s + 0.457·29-s − 0.981·31-s + 0.176·32-s − 0.594·34-s − 0.461·35-s − 1.55·37-s + 1.04·38-s + 0.115·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 + 7.26T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 + 8.19T + 89T^{2} \) |
| 97 | \( 1 - 6.73T + 97T^{2} \) |
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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
−8.791827720066956787144312068013, −7.72631866169123943631201932539, −6.95215657817845486665494552397, −6.36189585671489131292085073389, −5.37732281051159300999161447710, −4.89901461160539060648847913943, −3.45680866896735572553399382802, −3.06288125545901121212302253757, −1.90040085272481334899903322657, 0,
1.90040085272481334899903322657, 3.06288125545901121212302253757, 3.45680866896735572553399382802, 4.89901461160539060648847913943, 5.37732281051159300999161447710, 6.36189585671489131292085073389, 6.95215657817845486665494552397, 7.72631866169123943631201932539, 8.791827720066956787144312068013
