| L(s) = 1 | + (0.221 + 1.39i)2-s + (−1.08 + 1.34i)3-s + (−1.90 + 0.618i)4-s + (0.951 + 1.30i)5-s + (−2.12 − 1.22i)6-s + (−0.224 + 0.0729i)7-s + (−1.28 − 2.52i)8-s + (−0.633 − 2.93i)9-s + (−1.61 + 1.61i)10-s + (−2.19 + 2.48i)11-s + (1.23 − 3.23i)12-s + (0.809 + 0.587i)13-s + (−0.151 − 0.297i)14-s + (−2.79 − 0.142i)15-s + (3.23 − 2.35i)16-s + (2.99 + 4.11i)17-s + ⋯ |
| L(s) = 1 | + (0.156 + 0.987i)2-s + (−0.628 + 0.778i)3-s + (−0.951 + 0.309i)4-s + (0.425 + 0.585i)5-s + (−0.866 − 0.498i)6-s + (−0.0848 + 0.0275i)7-s + (−0.453 − 0.891i)8-s + (−0.211 − 0.977i)9-s + (−0.511 + 0.511i)10-s + (−0.660 + 0.750i)11-s + (0.356 − 0.934i)12-s + (0.224 + 0.163i)13-s + (−0.0405 − 0.0795i)14-s + (−0.722 − 0.0366i)15-s + (0.809 − 0.587i)16-s + (0.725 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.189367 + 0.858109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189367 + 0.858109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.221 - 1.39i)T \) |
| 3 | \( 1 + (1.08 - 1.34i)T \) |
| 11 | \( 1 + (2.19 - 2.48i)T \) |
| good | 5 | \( 1 + (-0.951 - 1.30i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.224 - 0.0729i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 4.11i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.61 - 1.5i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 + (3.80 - 1.23i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.44 + 4.73i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.16 - 9.73i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (9.00 + 2.92i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.23iT - 43T^{2} \) |
| 47 | \( 1 + (-2.04 + 6.29i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.96 + 9.59i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.263 + 0.812i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.92 + 3.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 2.85iT - 67T^{2} \) |
| 71 | \( 1 + (-4.92 + 3.57i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.38 + 7.33i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.0 - 13.8i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.66 - 2.66i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 7.18iT - 89T^{2} \) |
| 97 | \( 1 + (2.69 + 1.95i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−13.96953779760131311427344002202, −12.83746889192866727136740975396, −11.74185822202503826354632846064, −10.23037138142357252588999968238, −9.799756485850089585465377719825, −8.344515811302901444809187274541, −6.99404180732065999020852826115, −5.94715549967451401444960905448, −4.97931462118911426061935781230, −3.53221815942399785795183815205,
1.05259738368063356776793761515, 2.92889270768944333247036844439, 5.04524255292312620834835490156, 5.74037749632882784567519015047, 7.52404453197272753689872302404, 8.788061222700258753968690862406, 9.925537952051291046553235142058, 11.07866856361090818952157238962, 11.80392232152355251933714695732, 12.88644317970932744361819768717
