L(s) = 1 | + (1.52 + 1.29i)2-s + (0.637 + 3.94i)4-s + (−4.27 + 2.59i)5-s + 0.837·7-s + (−4.14 + 6.83i)8-s + (−9.87 − 1.59i)10-s + 15.7i·11-s + 5.18i·13-s + (1.27 + 1.08i)14-s + (−15.1 + 5.03i)16-s − 27.3i·17-s + 17.9i·19-s + (−12.9 − 15.2i)20-s + (−20.4 + 24.0i)22-s + 19.1·23-s + ⋯ |
L(s) = 1 | + (0.761 + 0.648i)2-s + (0.159 + 0.987i)4-s + (−0.854 + 0.518i)5-s + 0.119·7-s + (−0.518 + 0.854i)8-s + (−0.987 − 0.159i)10-s + 1.43i·11-s + 0.398i·13-s + (0.0910 + 0.0775i)14-s + (−0.949 + 0.314i)16-s − 1.60i·17-s + 0.945i·19-s + (−0.648 − 0.761i)20-s + (−0.930 + 1.09i)22-s + 0.830·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.748172 + 1.61964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.748172 + 1.61964i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.52 - 1.29i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.27 - 2.59i)T \) |
good | 7 | \( 1 - 0.837T + 49T^{2} \) |
| 11 | \( 1 - 15.7iT - 121T^{2} \) |
| 13 | \( 1 - 5.18iT - 169T^{2} \) |
| 17 | \( 1 + 27.3iT - 289T^{2} \) |
| 19 | \( 1 - 17.9iT - 361T^{2} \) |
| 23 | \( 1 - 19.1T + 529T^{2} \) |
| 29 | \( 1 - 45.6T + 841T^{2} \) |
| 31 | \( 1 - 13.6iT - 961T^{2} \) |
| 37 | \( 1 + 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 15.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 87.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 54.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 59.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 39.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 168. iT - 9.40e3T^{2} \) |
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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
−12.65093529329051323846269584666, −12.03646369982062248838871883104, −11.15398259791927374100888228759, −9.757138917286647718668356099017, −8.387708133826134010688676077935, −7.29480656905864131084920677238, −6.77748269608075191580155358129, −5.08465120676890535342822308687, −4.15018697163166027761421982132, −2.72500804699581193060984271347,
0.885931110136733617412145854802, 3.04249963774564307185595012233, 4.16880497506865574118289681564, 5.35443821781891416711532013046, 6.53106182403529467264691414485, 8.139263152582364598090650874327, 9.001558057102161683889985198825, 10.54864158624909902841076186675, 11.16805011745389464187734154297, 12.11234643926914395499908307533