L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 1.14i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.573 + 0.993i)10-s + (−3.84 + 2.22i)11-s + (−3.54 − 0.666i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.35 + 2.34i)17-s + (5.68 + 3.28i)19-s + (0.993 + 0.573i)20-s + (−2.22 + 3.84i)22-s + (1.03 + 1.80i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.513i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.181 + 0.314i)10-s + (−1.16 + 0.670i)11-s + (−0.982 − 0.184i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.328 + 0.569i)17-s + (1.30 + 0.752i)19-s + (0.222 + 0.128i)20-s + (−0.473 + 0.820i)22-s + (0.216 + 0.375i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.501977400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.501977400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (3.54 + 0.666i)T \) |
good | 5 | \( 1 - 1.14iT - 5T^{2} \) |
| 11 | \( 1 + (3.84 - 2.22i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.68 - 3.28i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.03 - 1.80i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.59 - 6.23i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.90iT - 31T^{2} \) |
| 37 | \( 1 + (8.35 - 4.82i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.22 + 4.74i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.70 - 2.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.67iT - 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + (0.0586 + 0.0338i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.05 - 7.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.444 - 0.256i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.34 + 5.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 8.02iT - 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 15.3iT - 83T^{2} \) |
| 89 | \( 1 + (9.40 - 5.42i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.84 - 1.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.848777662180376987647422672412, −8.881760611411587958328129414420, −7.66114123560151035196146004575, −7.19055855083400127946429050085, −6.30018151005825227793457186882, −5.19071533604658852582583959620, −4.76205886921106686727842884291, −3.33151598345395100137237007469, −2.86084337402300595353017255172, −1.56385712193918364956182058835,
0.44772477620575003666346420125, 2.44773306211082300296879890891, 3.07674329524338410004465084675, 4.44560856997809434411022835240, 5.09518694053969568151115210336, 5.76367918692382597932642295912, 6.79714520526197823318391228082, 7.56170213099036195814146414923, 8.258625041493479811035079664192, 9.222753993185920884183428055263