L(s) = 1 | + (1.37 + 0.331i)2-s + (1.78 + 0.910i)4-s + (−1.64 + 1.51i)5-s + 0.936·7-s + (2.14 + 1.84i)8-s + (−2.76 + 1.53i)10-s + 2.20i·11-s + 3.33·13-s + (1.28 + 0.310i)14-s + (2.34 + 3.24i)16-s + 1.54·17-s − 3.12·19-s + (−4.31 + 1.18i)20-s + (−0.731 + 3.03i)22-s − 3.39i·23-s + ⋯ |
L(s) = 1 | + (0.972 + 0.234i)2-s + (0.890 + 0.455i)4-s + (−0.737 + 0.675i)5-s + 0.353·7-s + (0.759 + 0.650i)8-s + (−0.875 + 0.483i)10-s + 0.665i·11-s + 0.924·13-s + (0.344 + 0.0828i)14-s + (0.585 + 0.810i)16-s + 0.374·17-s − 0.716·19-s + (−0.964 + 0.265i)20-s + (−0.155 + 0.647i)22-s − 0.707i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98421 + 1.06846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98421 + 1.06846i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.331i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.64 - 1.51i)T \) |
good | 7 | \( 1 - 0.936T + 7T^{2} \) |
| 11 | \( 1 - 2.20iT - 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 3.39iT - 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 + 8.30iT - 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 + 7.77iT - 43T^{2} \) |
| 47 | \( 1 + 10.7iT - 47T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 5.59iT - 73T^{2} \) |
| 79 | \( 1 + 1.02iT - 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.18iT - 97T^{2} \) |
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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
−11.47109841366005382878525144211, −11.12403885254121435376535049030, −10.01607612016556304227008177140, −8.422528100542854746761570344501, −7.64376182909417143458666663819, −6.74244325842205137313277880709, −5.78203743460949883643424025717, −4.42476589940222045110697256971, −3.65488936334108017365285847373, −2.23813132360046512826926803771,
1.39291943416209471853792107135, 3.28125474029047298270008617287, 4.18578352551049600790610189051, 5.26887277553601516930729720046, 6.20187480357381414183095378793, 7.50487623642488518825777849361, 8.364271420416786057640585457502, 9.483629031152633709159284961512, 11.02512794757933762285152327848, 11.18132813673209424418015582255