| L(s) = 1 | + 4.05·2-s − 1.73·3-s + 8.46·4-s − 5·5-s − 7.04·6-s − 13.5·7-s + 1.89·8-s − 23.9·9-s − 20.2·10-s − 11·11-s − 14.7·12-s + 56.9·13-s − 54.9·14-s + 8.68·15-s − 60.0·16-s + 76.4·17-s − 97.3·18-s − 19·19-s − 42.3·20-s + 23.5·21-s − 44.6·22-s + 108.·23-s − 3.29·24-s + 25·25-s + 231.·26-s + 88.5·27-s − 114.·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.334·3-s + 1.05·4-s − 0.447·5-s − 0.479·6-s − 0.730·7-s + 0.0837·8-s − 0.888·9-s − 0.641·10-s − 0.301·11-s − 0.353·12-s + 1.21·13-s − 1.04·14-s + 0.149·15-s − 0.938·16-s + 1.09·17-s − 1.27·18-s − 0.229·19-s − 0.473·20-s + 0.244·21-s − 0.432·22-s + 0.985·23-s − 0.0279·24-s + 0.200·25-s + 1.74·26-s + 0.631·27-s − 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.951297102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951297102\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 4.05T + 8T^{2} \) |
| 3 | \( 1 + 1.73T + 27T^{2} \) |
| 7 | \( 1 + 13.5T + 343T^{2} \) |
| 13 | \( 1 - 56.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 86.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 299.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 457.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 275.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 903.T + 9.12e5T^{2} \) |
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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
−9.504525033943667841458178558369, −8.712264944595502744796551885877, −7.68218464373452618802737486220, −6.59312190135511126875261765444, −5.89994604255358807515822013550, −5.32932502897196720516604813879, −4.19272183062776670449914410937, −3.38318888347350218292637352388, −2.68652320252151313808817105521, −0.72961454859138166686504889695,
0.72961454859138166686504889695, 2.68652320252151313808817105521, 3.38318888347350218292637352388, 4.19272183062776670449914410937, 5.32932502897196720516604813879, 5.89994604255358807515822013550, 6.59312190135511126875261765444, 7.68218464373452618802737486220, 8.712264944595502744796551885877, 9.504525033943667841458178558369
