L(s) = 1 | − 1.41·2-s − 2.41·3-s + 2.00·4-s + 9.43i·5-s + 3.41·6-s + 3.91i·7-s − 2.82·8-s − 3.17·9-s − 13.3i·10-s − 3.91i·11-s − 4.82·12-s + 10.3·13-s − 5.52i·14-s − 22.7i·15-s + 4.00·16-s + 17.2i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.804·3-s + 0.500·4-s + 1.88i·5-s + 0.569·6-s + 0.558i·7-s − 0.353·8-s − 0.352·9-s − 1.33i·10-s − 0.355i·11-s − 0.402·12-s + 0.793·13-s − 0.394i·14-s − 1.51i·15-s + 0.250·16-s + 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.377616 + 0.432468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.377616 + 0.432468i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 23 | \( 1 + (-3.10 - 22.7i)T \) |
good | 3 | \( 1 + 2.41T + 9T^{2} \) |
| 5 | \( 1 - 9.43iT - 25T^{2} \) |
| 7 | \( 1 - 3.91iT - 49T^{2} \) |
| 11 | \( 1 + 3.91iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 - 17.2iT - 289T^{2} \) |
| 19 | \( 1 + 26.7iT - 361T^{2} \) |
| 29 | \( 1 - 35.1T + 841T^{2} \) |
| 31 | \( 1 - 33.2T + 961T^{2} \) |
| 37 | \( 1 - 39.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 29.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 61.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 26.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 15.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 68.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 111.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 74.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 39.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.0iT - 9.40e3T^{2} \) |
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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
−15.79634774156744252501946737299, −15.02942744214106609055698228154, −13.70969222183042781531100051654, −11.71366836047391032886906754638, −11.10193417456454649209347536322, −10.19461390118246877246835072667, −8.465758620865359697609002984044, −6.79974019905306720134474615429, −5.96848964544129619481849641499, −2.94822052612872849515975147734,
0.875233081579683532506493433869, 4.65944039850875821033771103120, 6.07456471755279168268739410981, 8.007864450648612350181813417216, 9.038459394996796406323972988272, 10.37866245584566159990378862889, 11.78824140563728755205754292621, 12.54291888259145340438349260179, 13.95109105006546355236764767419, 15.97181319817367299070708736186