L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (0.853 + 6.94i)7-s + 2.82·8-s − 2.99·9-s + 2.88·11-s − 3.46i·12-s − 13.8i·13-s + (1.20 + 9.82i)14-s + 4.00·16-s + 24.1i·17-s − 4.24·18-s + 6.53i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.121 + 0.992i)7-s + 0.353·8-s − 0.333·9-s + 0.261·11-s − 0.288i·12-s − 1.06i·13-s + (0.0861 + 0.701i)14-s + 0.250·16-s + 1.42i·17-s − 0.235·18-s + 0.343i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.194537629\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.194537629\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.853 - 6.94i)T \) |
good | 11 | \( 1 - 2.88T + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 - 24.1iT - 289T^{2} \) |
| 19 | \( 1 - 6.53iT - 361T^{2} \) |
| 23 | \( 1 - 28.8T + 529T^{2} \) |
| 29 | \( 1 - 32.9T + 841T^{2} \) |
| 31 | \( 1 - 2.43iT - 961T^{2} \) |
| 37 | \( 1 - 50.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 120.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 21.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 78.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 44.1iT - 9.40e3T^{2} \) |
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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
−9.816476552199578528242316586118, −8.608082779417859172081043545515, −8.145979249620987237545000224168, −7.07927223809592867581793438535, −6.11751460816205523785188869657, −5.62316461568368475069672308642, −4.57165321641677291367542774596, −3.30302378784278220076602761500, −2.44827555866032364155504977619, −1.19860093363250882577272438286,
0.922608120316133728753286730212, 2.55188867130755533090920334488, 3.59239908751648403528186197343, 4.56611243621300946384831856931, 4.98207326216947178520325388149, 6.38556852416665885520331375112, 6.99909421882355482777458305775, 7.87618453901002464371615363906, 9.130022513295623862195420891645, 9.659115780589879026948262051801