L(s) = 1 | + (0.504 − 1.32i)2-s + (2.22 − 2.22i)3-s + (−1.49 − 1.33i)4-s + (0.707 + 0.707i)5-s + (−1.81 − 4.05i)6-s + i·7-s + (−2.51 + 1.29i)8-s − 6.85i·9-s + (1.29 − 0.577i)10-s + (−1.33 − 1.33i)11-s + (−6.26 + 0.349i)12-s + (0.657 − 0.657i)13-s + (1.32 + 0.504i)14-s + 3.13·15-s + (0.444 + 3.97i)16-s − 1.24·17-s + ⋯ |
L(s) = 1 | + (0.356 − 0.934i)2-s + (1.28 − 1.28i)3-s + (−0.745 − 0.666i)4-s + (0.316 + 0.316i)5-s + (−0.740 − 1.65i)6-s + 0.377i·7-s + (−0.888 + 0.458i)8-s − 2.28i·9-s + (0.408 − 0.182i)10-s + (−0.401 − 0.401i)11-s + (−1.80 + 0.100i)12-s + (0.182 − 0.182i)13-s + (0.353 + 0.134i)14-s + 0.810·15-s + (0.111 + 0.993i)16-s − 0.302·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.599573 - 2.32825i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.599573 - 2.32825i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.504 + 1.32i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-2.22 + 2.22i)T - 3iT^{2} \) |
| 11 | \( 1 + (1.33 + 1.33i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.657 + 0.657i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + (1.56 - 1.56i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.20iT - 23T^{2} \) |
| 29 | \( 1 + (-3.83 + 3.83i)T - 29iT^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 + (-4.06 - 4.06i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.5iT - 41T^{2} \) |
| 43 | \( 1 + (2.18 + 2.18i)T + 43iT^{2} \) |
| 47 | \( 1 + 1.62T + 47T^{2} \) |
| 53 | \( 1 + (-7.72 - 7.72i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.930 + 0.930i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.14 - 6.14i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.05 + 8.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 + 9.90T + 79T^{2} \) |
| 83 | \( 1 + (-4.56 + 4.56i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
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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
−10.32042627676324430516783841988, −9.553469853419027029444838782631, −8.438682864005210861752652816176, −8.163450732827378526600706651726, −6.64599272059076993842019114526, −5.97765058050954290736858870534, −4.36840681215624747293820416489, −2.92664246744911173407268758774, −2.54469580372980406677274724027, −1.19454711193223037495632979402,
2.56243538723442894171561735638, 3.72871795028270900571359011319, 4.54396136868306742692265780751, 5.26813568386923999590034689247, 6.68629258611163958078129495327, 7.79821347480436788369835474558, 8.504399519093641701460098894764, 9.222589278935870737896257354778, 9.927500881963547044588273675567, 10.75798019108121678365172665267