L(s) = 1 | + 1.39i·2-s − 1.73·3-s + 2.04·4-s + 0.273·5-s − 2.42i·6-s − 10.8·7-s + 8.45i·8-s + 2.99·9-s + 0.382i·10-s + 15.0i·11-s − 3.53·12-s + 11.1i·13-s − 15.2i·14-s − 0.473·15-s − 3.67·16-s − 3.40·17-s + ⋯ |
L(s) = 1 | + 0.699i·2-s − 0.577·3-s + 0.510·4-s + 0.0546·5-s − 0.404i·6-s − 1.55·7-s + 1.05i·8-s + 0.333·9-s + 0.0382i·10-s + 1.36i·11-s − 0.294·12-s + 0.854i·13-s − 1.08i·14-s − 0.0315·15-s − 0.229·16-s − 0.200·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.212145 + 0.889357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.212145 + 0.889357i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 59 | \( 1 + (-52.6 - 26.6i)T \) |
good | 2 | \( 1 - 1.39iT - 4T^{2} \) |
| 5 | \( 1 - 0.273T + 25T^{2} \) |
| 7 | \( 1 + 10.8T + 49T^{2} \) |
| 11 | \( 1 - 15.0iT - 121T^{2} \) |
| 13 | \( 1 - 11.1iT - 169T^{2} \) |
| 17 | \( 1 + 3.40T + 289T^{2} \) |
| 19 | \( 1 + 15.7T + 361T^{2} \) |
| 23 | \( 1 + 15.2iT - 529T^{2} \) |
| 29 | \( 1 - 22.5T + 841T^{2} \) |
| 31 | \( 1 - 59.5iT - 961T^{2} \) |
| 37 | \( 1 + 42.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 78.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 78.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 37.2T + 2.80e3T^{2} \) |
| 61 | \( 1 - 44.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 17.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 85.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 62.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 98.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 35.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 11.0iT - 9.40e3T^{2} \) |
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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
−12.58257779690237574070673029862, −12.17845632408353368225434463814, −10.80464523404307579405562453891, −9.950851019208396845051464985360, −8.835741487013820617554500818646, −7.16614224725934258955610444292, −6.73096032392147728084805305949, −5.75657610216674086005574869841, −4.25123671688110568579732737773, −2.32402830146647060804930381053,
0.55100635932318127334360703787, 2.74405032590947874509790643102, 3.80921165818730600022347011305, 5.95352633347708786964406351127, 6.38146116138623002885814171602, 7.84513087412966745116953434338, 9.443934788317760944323590775911, 10.21721881388114171792829577040, 11.10511316413289622243196151001, 11.89794614160565497883313072138