L(s) = 1 | − 2.55·2-s + 4.50·4-s + (1.39 + 2.41i)5-s + (−2.46 + 0.968i)7-s − 6.39·8-s + (−3.55 − 6.15i)10-s + (−1.38 − 2.39i)11-s + (2.99 − 2.01i)13-s + (6.28 − 2.46i)14-s + 7.28·16-s + 5.88·17-s + (−1.70 + 2.94i)19-s + (6.27 + 10.8i)20-s + (3.52 + 6.11i)22-s + 7.34·23-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 2.25·4-s + (0.623 + 1.07i)5-s + (−0.930 + 0.365i)7-s − 2.25·8-s + (−1.12 − 1.94i)10-s + (−0.417 − 0.722i)11-s + (0.830 − 0.557i)13-s + (1.67 − 0.659i)14-s + 1.82·16-s + 1.42·17-s + (−0.390 + 0.675i)19-s + (1.40 + 2.43i)20-s + (0.752 + 1.30i)22-s + 1.53·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.561713 + 0.367250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.561713 + 0.367250i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.46 - 0.968i)T \) |
| 13 | \( 1 + (-2.99 + 2.01i)T \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 + (-1.39 - 2.41i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.38 + 2.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 + (1.70 - 2.94i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 + (1.56 - 2.70i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.93 + 3.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 + (3.24 - 5.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.99 - 5.18i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.95 + 6.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.34 - 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 + (4.77 - 8.27i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.34 - 7.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.57 - 2.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.80 - 8.31i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 + 3.25i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + (-8.48 - 14.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.20150802389144540562567076633, −9.681199942992456757691497794671, −8.781790002029147908248677756495, −7.993991280474946150781231703684, −7.13931049880646931660687220685, −6.21146634130955336348450682550, −5.78512547725808455985628866975, −3.21663091253788526355191304686, −2.74412336188989428172654756858, −1.13514198303750923600903580682,
0.72485473202671129580102995660, 1.78034405490905666421876820743, 3.17699638571402430324718340857, 4.84622153604359856750109892077, 6.05383097829936017760803138700, 6.88541413347777353373055929110, 7.71155480561139933096636256720, 8.667801899538227400142046620519, 9.312421149594459314223868552872, 9.758991843276110003276828556553