| L(s) = 1 | + (0.643 − 0.152i)2-s + (−1.39 + 0.701i)4-s + (1.50 + 2.01i)5-s + (−3.43 + 2.25i)7-s + (−1.80 + 1.51i)8-s + (1.27 + 1.06i)10-s + (2.94 + 0.343i)11-s + (0.930 + 3.10i)13-s + (−1.86 + 1.97i)14-s + (0.935 − 1.25i)16-s + (0.572 − 3.24i)17-s + (−0.571 − 3.23i)19-s + (−3.50 − 1.76i)20-s + (1.94 − 0.227i)22-s + (2.08 + 1.37i)23-s + ⋯ |
| L(s) = 1 | + (0.455 − 0.107i)2-s + (−0.698 + 0.350i)4-s + (0.671 + 0.901i)5-s + (−1.29 + 0.854i)7-s + (−0.638 + 0.535i)8-s + (0.402 + 0.337i)10-s + (0.886 + 0.103i)11-s + (0.257 + 0.861i)13-s + (−0.498 + 0.528i)14-s + (0.233 − 0.314i)16-s + (0.138 − 0.787i)17-s + (−0.131 − 0.743i)19-s + (−0.784 − 0.394i)20-s + (0.414 − 0.0484i)22-s + (0.434 + 0.286i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.935127 + 0.816224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935127 + 0.816224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.643 + 0.152i)T + (1.78 - 0.897i)T^{2} \) |
| 5 | \( 1 + (-1.50 - 2.01i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (3.43 - 2.25i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (-2.94 - 0.343i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.930 - 3.10i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.572 + 3.24i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.571 + 3.23i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.08 - 1.37i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-3.82 - 4.04i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (0.373 - 6.41i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (2.56 - 0.935i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-8.80 - 2.08i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (3.09 + 7.17i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.588 + 10.1i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (0.00494 - 0.00856i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (11.2 - 1.32i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-8.05 - 4.04i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (4.01 - 4.25i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-1.81 - 1.52i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.61 + 3.03i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.75 + 2.07i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-3.60 + 0.853i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-9.57 + 8.03i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (4.27 - 5.73i)T + (-27.8 - 92.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.34444539648550087218032373911, −11.67054070521229825830500379481, −10.30615502239917924573674377216, −9.206925521275240846045725949665, −8.942831436052927238282455735591, −6.95357460053370953904502238019, −6.31540206830807499640207355119, −5.08153649634329676970098023394, −3.56308876114829317515419742956, −2.63368414103916982819385174730,
0.947549244488307460894798128456, 3.49593224645809101905197570254, 4.45438463880115663595417433750, 5.84759598099634557749329929539, 6.37507696205456423605731011195, 8.069364561158330204046179298694, 9.288534955063164032051387772797, 9.738403254722913591933813656699, 10.71201334143429227510406907317, 12.47433274062066984736302898620
