L(s) = 1 | + (1.85 + 1.35i)2-s + (1.01 + 3.12i)4-s + (1.37 − 0.997i)5-s + (0.0641 + 0.197i)7-s + (−0.910 + 2.80i)8-s + 3.90·10-s + (−1.23 + 3.07i)11-s + (−1.11 − 0.812i)13-s + (−0.147 + 0.453i)14-s + (−0.168 + 0.122i)16-s + (−2.97 + 2.15i)17-s + (2.50 − 7.70i)19-s + (4.50 + 3.27i)20-s + (−6.45 + 4.05i)22-s − 3.99·23-s + ⋯ |
L(s) = 1 | + (1.31 + 0.955i)2-s + (0.507 + 1.56i)4-s + (0.613 − 0.446i)5-s + (0.0242 + 0.0746i)7-s + (−0.321 + 0.990i)8-s + 1.23·10-s + (−0.372 + 0.928i)11-s + (−0.310 − 0.225i)13-s + (−0.0394 + 0.121i)14-s + (−0.0420 + 0.0305i)16-s + (−0.720 + 0.523i)17-s + (0.574 − 1.76i)19-s + (1.00 + 0.732i)20-s + (−1.37 + 0.864i)22-s − 0.833·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15397 + 1.49449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15397 + 1.49449i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.23 - 3.07i)T \) |
good | 2 | \( 1 + (-1.85 - 1.35i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.37 + 0.997i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.0641 - 0.197i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (1.11 + 0.812i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.97 - 2.15i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.50 + 7.70i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + (0.167 + 0.515i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.95 + 5.77i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.381 + 1.17i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.88 - 8.87i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 + (-2.42 + 7.47i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.30 - 3.85i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.43 + 7.48i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 0.816i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 5.85T + 67T^{2} \) |
| 71 | \( 1 + (10.3 - 7.49i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.36 - 7.27i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.69 - 3.41i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (11.2 - 8.15i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 6.19T + 89T^{2} \) |
| 97 | \( 1 + (-5.68 - 4.12i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.38560308814472204513708015680, −11.31487771357301366388541934351, −9.958647325348700474653971076933, −9.009563229004637016417339189484, −7.65558976878979309286466422039, −6.91020327640847583930529814967, −5.75477804861109614053090449913, −5.02797489978215035364102751444, −4.05788206132539972428825187757, −2.39214183837543640507628896943,
1.90810117519678105774943179390, 3.07451244414256057759485608115, 4.14435678837940629000736164509, 5.49390809394920957370035289466, 6.08702211363083666682297537118, 7.54259832580056675642976539308, 8.980389194561914078370434353162, 10.29299603088405711762256628556, 10.68361083840570561824043847149, 11.79938846542205341743795416403