| L(s) = 1 | − 20.7i·5-s + 63.5·7-s − 42.4i·11-s − 235.·13-s + 207. i·17-s + 227.·19-s + 657. i·23-s + 194.·25-s + 615. i·29-s + 1.61e3·31-s − 1.31e3i·35-s + 1.48e3·37-s + 2.76e3i·41-s − 827.·43-s − 926. i·47-s + ⋯ |
| L(s) = 1 | − 0.830i·5-s + 1.29·7-s − 0.350i·11-s − 1.39·13-s + 0.718i·17-s + 0.630·19-s + 1.24i·23-s + 0.310·25-s + 0.731i·29-s + 1.67·31-s − 1.07i·35-s + 1.08·37-s + 1.64i·41-s − 0.447·43-s − 0.419i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.469815982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469815982\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 20.7iT - 625T^{2} \) |
| 7 | \( 1 - 63.5T + 2.40e3T^{2} \) |
| 11 | \( 1 + 42.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 235.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 207. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 227.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 657. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 615. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.61e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.48e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.76e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 827.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 926. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.92e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.94e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 946.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.68e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.02e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.52e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.54e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.07e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.38e4T + 8.85e7T^{2} \) |
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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
−9.590896831300344682430871755705, −8.610544472896935499516941449556, −8.004478941029368883656153736844, −7.24792467514301327301574742700, −5.93843927056175882445301755542, −4.94064747200785195934328117524, −4.58351243649719569568306666478, −3.10055451542932009640424466756, −1.77581150381249653611477313999, −0.894587825541325335710071421777,
0.70468593712833331661407832623, 2.19048576528218868140868749592, 2.86942540971683387632821173384, 4.47564123605508126933661703968, 4.94199513202966229267665648810, 6.19956414396325442565418616636, 7.23539414646172243067330873725, 7.69444915236499241780540448552, 8.688477996005813099431422208269, 9.766290890613117984269590542449
