L(s) = 1 | + (−0.561 − 0.827i)2-s + (−1.30 − 1.13i)3-s + (−0.370 + 0.928i)4-s + (−1.66 − 3.59i)5-s + (−0.209 + 1.71i)6-s + (1.10 − 3.97i)7-s + (0.976 − 0.214i)8-s + (0.410 + 2.97i)9-s + (−2.04 + 3.39i)10-s + (0.859 − 0.814i)11-s + (1.54 − 0.791i)12-s + (0.0212 + 0.195i)13-s + (−3.90 + 1.31i)14-s + (−1.91 + 6.58i)15-s + (−0.725 − 0.687i)16-s + (2.29 − 0.637i)17-s + ⋯ |
L(s) = 1 | + (−0.396 − 0.585i)2-s + (−0.753 − 0.656i)3-s + (−0.185 + 0.464i)4-s + (−0.743 − 1.60i)5-s + (−0.0853 + 0.701i)6-s + (0.416 − 1.50i)7-s + (0.345 − 0.0760i)8-s + (0.136 + 0.990i)9-s + (−0.645 + 1.07i)10-s + (0.259 − 0.245i)11-s + (0.444 − 0.228i)12-s + (0.00589 + 0.0541i)13-s + (−1.04 + 0.351i)14-s + (−0.495 + 1.69i)15-s + (−0.181 − 0.171i)16-s + (0.556 − 0.154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.177607 + 0.553672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.177607 + 0.553672i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.561 + 0.827i)T \) |
| 3 | \( 1 + (1.30 + 1.13i)T \) |
| 59 | \( 1 + (7.67 + 0.244i)T \) |
good | 5 | \( 1 + (1.66 + 3.59i)T + (-3.23 + 3.81i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 3.97i)T + (-5.99 - 3.60i)T^{2} \) |
| 11 | \( 1 + (-0.859 + 0.814i)T + (0.595 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0212 - 0.195i)T + (-12.6 + 2.79i)T^{2} \) |
| 17 | \( 1 + (-2.29 + 0.637i)T + (14.5 - 8.76i)T^{2} \) |
| 19 | \( 1 + (4.88 - 3.71i)T + (5.08 - 18.3i)T^{2} \) |
| 23 | \( 1 + (0.939 - 1.77i)T + (-12.9 - 19.0i)T^{2} \) |
| 29 | \( 1 + (2.66 + 1.80i)T + (10.7 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-5.35 + 7.04i)T + (-8.29 - 29.8i)T^{2} \) |
| 37 | \( 1 + (1.22 - 5.55i)T + (-33.5 - 15.5i)T^{2} \) |
| 41 | \( 1 + (-2.22 + 1.18i)T + (23.0 - 33.9i)T^{2} \) |
| 43 | \( 1 + (-3.65 + 3.86i)T + (-2.32 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-10.8 - 5.03i)T + (30.4 + 35.8i)T^{2} \) |
| 53 | \( 1 + (-3.60 - 5.99i)T + (-24.8 + 46.8i)T^{2} \) |
| 61 | \( 1 + (7.38 - 5.00i)T + (22.5 - 56.6i)T^{2} \) |
| 67 | \( 1 + (1.71 + 7.79i)T + (-60.8 + 28.1i)T^{2} \) |
| 71 | \( 1 + (-6.13 + 13.2i)T + (-45.9 - 54.1i)T^{2} \) |
| 73 | \( 1 + (2.79 + 8.28i)T + (-58.1 + 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.0578 - 1.06i)T + (-78.5 + 8.54i)T^{2} \) |
| 83 | \( 1 + (2.01 + 12.2i)T + (-78.6 + 26.5i)T^{2} \) |
| 89 | \( 1 + (4.74 - 7.00i)T + (-32.9 - 82.6i)T^{2} \) |
| 97 | \( 1 + (-2.86 + 8.51i)T + (-77.2 - 58.7i)T^{2} \) |
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
−11.02663710952367385844796677082, −10.24332472604186791797974963458, −8.997635977689379929908964226189, −7.83266352901242557135713574377, −7.64738593058297270205366200782, −6.00322848857636626462407652442, −4.60967502182373618364703518745, −4.02297300699470134093344970061, −1.50526507853068444599227013640, −0.53830814956271407452846260890,
2.64881878386108398271615726657, 4.11658849899374550970703184324, 5.40261927656647038569417634806, 6.34261363900586171476233532234, 7.07458286034570352523291924009, 8.324590422232054358777442244861, 9.217223781621695077616072102413, 10.36766505625515520752522759753, 10.95874071445112054102569648763, 11.76572061142841020424065648280