L(s) = 1 | + (−4.98 − 0.385i)5-s + (−6.51 − 3.76i)7-s + (5.77 + 3.33i)11-s + (13.4 − 7.78i)13-s + 18.9·17-s − 27.1·19-s + (−9.80 − 16.9i)23-s + (24.7 + 3.84i)25-s + (−6.03 − 3.48i)29-s + (−14.7 − 25.5i)31-s + (31.0 + 21.2i)35-s + 52.2i·37-s + (52.7 − 30.4i)41-s + (−9.23 − 5.33i)43-s + (37.9 − 65.6i)47-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0770i)5-s + (−0.931 − 0.537i)7-s + (0.525 + 0.303i)11-s + (1.03 − 0.598i)13-s + 1.11·17-s − 1.42·19-s + (−0.426 − 0.738i)23-s + (0.988 + 0.153i)25-s + (−0.208 − 0.120i)29-s + (−0.476 − 0.824i)31-s + (0.887 + 0.607i)35-s + 1.41i·37-s + (1.28 − 0.742i)41-s + (−0.214 − 0.123i)43-s + (0.806 − 1.39i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.008543287635\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.008543287635\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.98 + 0.385i)T \) |
good | 7 | \( 1 + (6.51 + 3.76i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.77 - 3.33i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-13.4 + 7.78i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 19 | \( 1 + 27.1T + 361T^{2} \) |
| 23 | \( 1 + (9.80 + 16.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (6.03 + 3.48i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (14.7 + 25.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 52.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-52.7 + 30.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (9.23 + 5.33i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-37.9 + 65.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 47.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (21.1 - 12.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (20.3 - 35.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (51.0 - 29.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 33.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 79.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (43.0 - 74.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (76.5 - 132. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 83.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (14.0 + 8.10i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.576356917214831460838186459166, −8.576284563390445404059118337424, −8.050240411213954267928466305590, −7.10134452386494655429499105094, −6.43960789667473915433393286196, −5.55680182494032110024252432945, −4.11135174920118135695453502214, −3.87488660881016653913383881496, −2.76907877242658806714906640058, −1.10793844865530293424586712832,
0.00269586841878728884873324298, 1.48853711818420771353670069333, 3.01539101384430280575439466837, 3.69442916180155930610100578992, 4.47700004396070556455492771986, 5.91276848877520763267015794214, 6.28964933206448409856996699203, 7.33709703313901960241564080298, 8.082563334897370315882263642403, 9.010000026481992023753134485068