L(s) = 1 | + (0.988 + 3.87i)2-s + (−14.0 + 7.66i)4-s − 20.7·5-s − 5.38i·7-s + (−43.5 − 46.8i)8-s + (−20.5 − 80.4i)10-s − 115. i·11-s + 207.·13-s + (20.8 − 5.32i)14-s + (138. − 215. i)16-s − 383.·17-s − 618. i·19-s + (291. − 159. i)20-s + (448. − 114. i)22-s + 82.9i·23-s + ⋯ |
L(s) = 1 | + (0.247 + 0.968i)2-s + (−0.877 + 0.479i)4-s − 0.830·5-s − 0.109i·7-s + (−0.681 − 0.732i)8-s + (−0.205 − 0.804i)10-s − 0.955i·11-s + 1.22·13-s + (0.106 − 0.0271i)14-s + (0.540 − 0.841i)16-s − 1.32·17-s − 1.71i·19-s + (0.728 − 0.397i)20-s + (0.925 − 0.236i)22-s + 0.156i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.593095 - 0.351981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593095 - 0.351981i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.988 - 3.87i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 20.7T + 625T^{2} \) |
| 7 | \( 1 + 5.38iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 115. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 207.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 383.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 618. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 82.9iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 201.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 196. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 340.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.79e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 254. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.11e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.59e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.38e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.34e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.97e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.00e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.36e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.17e3T + 8.85e7T^{2} \) |
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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
−13.35269266253872786518548778861, −11.76224664648092130264049324039, −10.89666217146270925333368430209, −9.047286533595901091444605620355, −8.385604271397851155557630979406, −7.14285930456211267197797063165, −6.08980664148395274516994555388, −4.59148165629686441900472388283, −3.41761799747451218949114043043, −0.28950389968241938465870342288,
1.72864674593953113121492134860, 3.55987204461433798083148657217, 4.55406106794209802551637908215, 6.18293726609568318293296600654, 7.913724145986136936049132489944, 8.971953249916687759133277316368, 10.22037925119824875567009600086, 11.19360061402805580676405395863, 12.06972077963255669303893393022, 12.93084458955580812752452555304