L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−3.46 + 2i)7-s − 0.999i·8-s + (1.5 + 2.59i)11-s + (−3.46 − 2i)13-s + (−1.99 + 3.46i)14-s + (−0.5 − 0.866i)16-s + 3i·17-s + 4·19-s + (2.59 + 1.5i)22-s + (5.19 + 3i)23-s − 3.99·26-s + 3.99i·28-s + (3 + 5.19i)29-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.30 + 0.755i)7-s − 0.353i·8-s + (0.452 + 0.783i)11-s + (−0.960 − 0.554i)13-s + (−0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.727i·17-s + 0.917·19-s + (0.553 + 0.319i)22-s + (1.08 + 0.625i)23-s − 0.784·26-s + 0.755i·28-s + (0.557 + 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.570083639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570083639\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.46 + 2i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.3 - 6i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (6.06 - 3.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.661694263550316341077268300903, −9.348439282594727545308613741123, −8.150144666234174911310033726472, −6.96103248341772791580574854932, −6.53627746934021715804196715200, −5.38562547495512621390246873111, −4.82638874836616435354094745209, −3.36099241622914863781522462628, −2.96298453664751836794213367225, −1.54560008975365690710728840645,
0.52001905300213681786678967794, 2.54135215374031022065314848866, 3.44860247441931919091062349752, 4.24731318395897550072025837613, 5.30818309224393171443392242219, 6.20624551650071824098263319084, 7.03306373353913320430697912511, 7.40915079925229517464564549703, 8.704510711467057856403973119334, 9.507274221794895867145966053909