L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.346 + 0.199i)5-s + (−1.03 + 2.43i)7-s + 0.999·8-s + (−0.346 + 0.199i)10-s + (−2.70 + 1.91i)11-s + 0.164i·13-s + (−1.58 − 2.11i)14-s + (−0.5 + 0.866i)16-s + (0.906 + 1.57i)17-s + (5.41 + 3.12i)19-s − 0.399i·20-s + (−0.308 − 3.30i)22-s + (−3.76 − 2.17i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.154 + 0.0894i)5-s + (−0.392 + 0.919i)7-s + 0.353·8-s + (−0.109 + 0.0632i)10-s + (−0.815 + 0.578i)11-s + 0.0456i·13-s + (−0.424 − 0.565i)14-s + (−0.125 + 0.216i)16-s + (0.219 + 0.380i)17-s + (1.24 + 0.717i)19-s − 0.0894i·20-s + (−0.0656 − 0.704i)22-s + (−0.785 − 0.453i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4738710940\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4738710940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.03 - 2.43i)T \) |
| 11 | \( 1 + (2.70 - 1.91i)T \) |
good | 5 | \( 1 + (-0.346 - 0.199i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 0.164iT - 13T^{2} \) |
| 17 | \( 1 + (-0.906 - 1.57i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.41 - 3.12i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.76 + 2.17i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 + (0.141 + 0.245i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.40 + 4.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 2.07iT - 43T^{2} \) |
| 47 | \( 1 + (0.367 + 0.212i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.71 - 4.45i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.92 - 3.99i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.10 - 3.52i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0327 - 0.0567i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.43iT - 71T^{2} \) |
| 73 | \( 1 + (7.21 - 4.16i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.531 + 0.306i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.91T + 83T^{2} \) |
| 89 | \( 1 + (8.89 + 5.13i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−9.889188173646022527659054910150, −9.268741092746662149922673512851, −8.256119737945429607374747043155, −7.74787905685661425590278141387, −6.76683331296561057577473917291, −5.83676300576194686552261901313, −5.38546965995357320015095206621, −4.18638242111564396138872424950, −2.89866118824062001041495324952, −1.78910927064300440469014340658,
0.21296625449597242969488000439, 1.54884409924130399177976544285, 3.00035459092470203000082730104, 3.61111199739938079106778677713, 4.85155019657770103944139205787, 5.67234038931188777437722804105, 6.89138478392115332202752800983, 7.63648397582154726107823484697, 8.288323391325861110843684378162, 9.561248366608185478836514007739