| L(s) = 1 | − 2.55i·3-s + (2.71 − 4.19i)5-s + 1.70·7-s + 2.48·9-s − 5.03i·11-s − 17.6i·13-s + (−10.7 − 6.92i)15-s + 3.02·17-s + 31.5i·19-s − 4.34i·21-s + (22.7 − 3.18i)23-s + (−10.2 − 22.7i)25-s − 29.3i·27-s − 20.7·29-s + 18.6·31-s + ⋯ |
| L(s) = 1 | − 0.850i·3-s + (0.542 − 0.839i)5-s + 0.243·7-s + 0.276·9-s − 0.457i·11-s − 1.35i·13-s + (−0.714 − 0.461i)15-s + 0.177·17-s + 1.65i·19-s − 0.207i·21-s + (0.990 − 0.138i)23-s + (−0.410 − 0.911i)25-s − 1.08i·27-s − 0.716·29-s + 0.601·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.936954497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936954497\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.71 + 4.19i)T \) |
| 23 | \( 1 + (-22.7 + 3.18i)T \) |
| good | 3 | \( 1 + 2.55iT - 9T^{2} \) |
| 7 | \( 1 - 1.70T + 49T^{2} \) |
| 11 | \( 1 + 5.03iT - 121T^{2} \) |
| 13 | \( 1 + 17.6iT - 169T^{2} \) |
| 17 | \( 1 - 3.02T + 289T^{2} \) |
| 19 | \( 1 - 31.5iT - 361T^{2} \) |
| 29 | \( 1 + 20.7T + 841T^{2} \) |
| 31 | \( 1 - 18.6T + 961T^{2} \) |
| 37 | \( 1 - 2.81T + 1.36e3T^{2} \) |
| 41 | \( 1 + 73.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 54.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 34.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 17.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 52.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 21.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 69.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 37.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 48.8T + 9.40e3T^{2} \) |
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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
−10.38481911032665842925113362873, −9.790067688781189023106768099685, −8.397536886061520068288122132693, −8.055740630367931516695070602680, −6.84071723360293331920684436062, −5.79635123707367670544785430912, −5.02307133965921177778471856812, −3.49611043639578722653664519995, −1.89031736490050040728565513070, −0.844514098787062810284351796858,
1.84023264155369672161151974098, 3.19163271134354630273817342963, 4.42448575650957183103551416768, 5.21688282198594928858239158791, 6.74006121112023916188809196379, 7.11260352920678195395871764294, 8.718895029872007827533680565195, 9.554485689975853665670527893778, 10.09875406074534696681005620926, 11.13172616233858429154491733825
