| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s + 229.·7-s − 64·8-s + 81·9-s − 100·10-s − 291.·11-s + 144·12-s + 1.15e3·13-s − 919.·14-s + 225·15-s + 256·16-s + 1.16e3·17-s − 324·18-s + 361·19-s + 400·20-s + 2.06e3·21-s + 1.16e3·22-s − 3.80e3·23-s − 576·24-s + 625·25-s − 4.63e3·26-s + 729·27-s + 3.67e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.77·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.725·11-s + 0.288·12-s + 1.89·13-s − 1.25·14-s + 0.258·15-s + 0.250·16-s + 0.975·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 1.02·21-s + 0.512·22-s − 1.50·23-s − 0.204·24-s + 0.200·25-s − 1.34·26-s + 0.192·27-s + 0.886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.210742113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.210742113\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 229.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 291.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.16e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.68e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.85e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.28e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.32e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.78e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.20e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.15e4T + 8.58e9T^{2} \) |
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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
−10.06432391028400221728845347383, −8.815408916341289331041944040276, −8.142021917787282396233894005400, −7.84432285112648576979354763346, −6.39117350858204798412437045619, −5.43867897393753459748578897190, −4.27711143770156740478370330081, −2.93006488273541922985060124379, −1.72165015650152635579501601491, −1.07597534151201279847204857993,
1.07597534151201279847204857993, 1.72165015650152635579501601491, 2.93006488273541922985060124379, 4.27711143770156740478370330081, 5.43867897393753459748578897190, 6.39117350858204798412437045619, 7.84432285112648576979354763346, 8.142021917787282396233894005400, 8.815408916341289331041944040276, 10.06432391028400221728845347383
