L(s) = 1 | + (−1.27 + 0.620i)2-s − 0.701i·3-s + (1.22 − 1.57i)4-s + (−1.10 − 1.94i)5-s + (0.435 + 0.891i)6-s − 1.18·7-s + (−0.581 + 2.76i)8-s + 2.50·9-s + (2.61 + 1.78i)10-s − 1.72i·11-s + (−1.10 − 0.862i)12-s − 2.64·13-s + (1.50 − 0.736i)14-s + (−1.36 + 0.775i)15-s + (−0.980 − 3.87i)16-s − 4.62i·17-s + ⋯ |
L(s) = 1 | + (−0.898 + 0.439i)2-s − 0.405i·3-s + (0.614 − 0.789i)4-s + (−0.493 − 0.869i)5-s + (0.177 + 0.364i)6-s − 0.448·7-s + (−0.205 + 0.978i)8-s + 0.835·9-s + (0.825 + 0.564i)10-s − 0.518i·11-s + (−0.319 − 0.249i)12-s − 0.733·13-s + (0.402 − 0.196i)14-s + (−0.352 + 0.200i)15-s + (−0.245 − 0.969i)16-s − 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.227929 - 0.455427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.227929 - 0.455427i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.27 - 0.620i)T \) |
| 5 | \( 1 + (1.10 + 1.94i)T \) |
| 19 | \( 1 + (2.08 - 3.82i)T \) |
good | 3 | \( 1 + 0.701iT - 3T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 1.72iT - 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + 4.62iT - 17T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + 7.53iT - 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 + 7.40iT - 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 + 9.50iT - 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 - 6.47iT - 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 - 3.39T + 97T^{2} \) |
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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
−10.94226625593899019456783269155, −9.728307125568895402666381397678, −9.324983878296387812149283785748, −7.971380279836266605717249921562, −7.61851846688071521591441828552, −6.44220283704872081870876139239, −5.41308634644877358813607869198, −4.06257065373566808266853548353, −2.07344453887985277462457521134, −0.43595384325984886863961389191,
2.07810618224780969977287869584, 3.43831859692021025813807260972, 4.37501406323874440324284073121, 6.33330240891490209455115001224, 7.19513735830543685824891111840, 7.905769392383222986065258681436, 9.214841677920689640592696800107, 9.935241167518100033611492773244, 10.63459325694852449099448149872, 11.32309136674384656432867748190