L(s) = 1 | + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.139 − 1.85i)5-s + (0.0747 − 0.997i)7-s + (0.433 − 0.900i)8-s + (1.85 + 0.139i)10-s + (1.77 − 0.698i)11-s + (0.974 + 0.222i)14-s + (0.826 + 0.563i)16-s + (−0.414 + 1.81i)20-s + (0.425 + 1.86i)22-s + (−2.43 + 0.367i)25-s + (−0.365 + 0.930i)28-s + (1.92 + 0.440i)29-s + (−0.975 − 0.563i)31-s + (−0.680 + 0.733i)32-s + ⋯ |
L(s) = 1 | + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.139 − 1.85i)5-s + (0.0747 − 0.997i)7-s + (0.433 − 0.900i)8-s + (1.85 + 0.139i)10-s + (1.77 − 0.698i)11-s + (0.974 + 0.222i)14-s + (0.826 + 0.563i)16-s + (−0.414 + 1.81i)20-s + (0.425 + 1.86i)22-s + (−2.43 + 0.367i)25-s + (−0.365 + 0.930i)28-s + (1.92 + 0.440i)29-s + (−0.975 − 0.563i)31-s + (−0.680 + 0.733i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.129584121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.129584121\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.149 - 0.988i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.0747 + 0.997i)T \) |
good | 5 | \( 1 + (0.139 + 1.85i)T + (-0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (-1.77 + 0.698i)T + (0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (-1.92 - 0.440i)T + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.294 + 0.955i)T + (-0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.129 - 1.72i)T + (-0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.129 - 0.858i)T + (-0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.185 - 0.233i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 97 | \( 1 - 1.36iT - T^{2} \) |
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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
−8.567945748897220942588026989990, −8.058285599375813085418393409177, −7.14071399784702949103092593116, −6.44044123022627241406787356966, −5.64502215127928751854065935370, −4.82131414728922704376000024060, −4.18591086110746024164466140618, −3.67446448589380604119850954469, −1.37953422075523856297589078610, −0.825228992129413976653640403835,
1.64850857802746986346380436302, 2.46757663722861670735428961561, 3.22511313990971088702827254971, 3.92790779179895680913476911743, 4.84736851653499242856546930187, 6.06180346339444754696727810044, 6.61174657901261994408908702617, 7.38302554339866374146383681205, 8.285816892661315941934740092108, 9.100753694178095175010201537795