L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.292 − 0.507i)11-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + (−0.499 − 0.866i)18-s + (−1.41 + 2.44i)19-s − 0.999·20-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.0883 − 0.152i)11-s + (0.144 − 0.249i)12-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.171 + 0.297i)17-s + (−0.117 − 0.204i)18-s + (−0.324 + 0.561i)19-s − 0.223·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.434283733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434283733\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.41 - 4.18i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + (-2.53 - 4.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 + (4.53 - 7.85i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.65 - 8.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.24 - 2.15i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.82 - 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.94 + 6.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + (-3.82 - 6.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.65 + 9.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + (3.24 - 5.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−9.581961138115794969210438547327, −8.918263932218284489729966420701, −8.164345869420223441835790706996, −7.54976314312344354999832814189, −6.36220492138492771204754876556, −5.72044241218869125995973502928, −4.76387889411476678885022073650, −3.94977606435075788689933479281, −2.67844886011631871232655950755, −1.24186694914375611279756414302,
0.69896090537668797638094349520, 2.16471505500199443364508523404, 2.79852408529432256747290262419, 3.96594148434104622741039151844, 4.97797299996805279536919651508, 6.24379037902097920300892982671, 6.88467179314056365041275759683, 7.87542522264148192243774545077, 8.452690430784913540135374936805, 9.376759436596906222649768446029