| 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 | \( 1 + (-2.86 - 8.51i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−11.76572061142841020424065648280, −10.95874071445112054102569648763, −10.36766505625515520752522759753, −9.217223781621695077616072102413, −8.324590422232054358777442244861, −7.07458286034570352523291924009, −6.34261363900586171476233532234, −5.40261927656647038569417634806, −4.11658849899374550970703184324, −2.64881878386108398271615726657,
0.53830814956271407452846260890, 1.50526507853068444599227013640, 4.02297300699470134093344970061, 4.60967502182373618364703518745, 6.00322848857636626462407652442, 7.64738593058297270205366200782, 7.83266352901242557135713574377, 8.997635977689379929908964226189, 10.24332472604186791797974963458, 11.02663710952367385844796677082
