L(s) = 1 | + (1.47 − 0.658i)2-s + (0.413 − 0.459i)4-s + (−0.348 − 0.155i)5-s + (−2.93 − 0.623i)7-s + (−0.690 + 2.12i)8-s − 0.618·10-s + (−2.93 + 1.55i)11-s + (−0.651 + 6.20i)13-s + (−4.74 + 1.00i)14-s + (0.507 + 4.82i)16-s + (0.5 − 0.363i)17-s + (−0.263 + 0.812i)19-s + (−0.215 + 0.0960i)20-s + (−3.31 + 4.22i)22-s + (2.73 + 4.73i)23-s + ⋯ |
L(s) = 1 | + (1.04 − 0.465i)2-s + (0.206 − 0.229i)4-s + (−0.156 − 0.0694i)5-s + (−1.10 − 0.235i)7-s + (−0.244 + 0.751i)8-s − 0.195·10-s + (−0.883 + 0.467i)11-s + (−0.180 + 1.72i)13-s + (−1.26 + 0.269i)14-s + (0.126 + 1.20i)16-s + (0.121 − 0.0881i)17-s + (−0.0605 + 0.186i)19-s + (−0.0482 + 0.0214i)20-s + (−0.706 + 0.900i)22-s + (0.570 + 0.988i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.822375 + 0.933518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.822375 + 0.933518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (2.93 - 1.55i)T \) |
good | 2 | \( 1 + (-1.47 + 0.658i)T + (1.33 - 1.48i)T^{2} \) |
| 5 | \( 1 + (0.348 + 0.155i)T + (3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (2.93 + 0.623i)T + (6.39 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.651 - 6.20i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.363i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.73 - 4.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.37 - 0.929i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.402 + 3.83i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.30 + 4.02i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (5.81 - 1.23i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.881 - 1.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.413 + 0.459i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-5.97 - 4.33i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.56 - 3.95i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.119 + 1.13i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (5.28 + 9.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.381 - 1.17i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.482 + 0.214i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.32 - 12.6i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 + (-13.7 + 6.11i)T + (64.9 - 72.0i)T^{2} \) |
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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
−10.41889294855106526638794933387, −9.626465654640746867830248193310, −8.810040260157584214525180590490, −7.68174679378384401196551899784, −6.76977438701807026798702035200, −5.85486998189335735310230057749, −4.78933666943923278635311387873, −4.05639466576047948495721223216, −3.11717240171267703815137390953, −2.06901913463720509589744936467,
0.39856015733096538977455274397, 2.95368857069207500490553582026, 3.33948699796223001229033983555, 4.76373621016924875295108260456, 5.50930275290254462460741915513, 6.21623874184910495555170700385, 7.08057126455311528461881961176, 8.062098997750088715350348830189, 9.010959551724332940700656237963, 10.20791309404617106899376096329