L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (2.23 + 0.0345i)5-s + (0.796 + 0.796i)7-s + (0.707 + 0.707i)8-s + (−1.60 + 1.55i)10-s − 2.14i·11-s + (3.95 − 3.95i)13-s − 1.12·14-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 4.73i·19-s + (0.0345 − 2.23i)20-s + (1.51 + 1.51i)22-s + (−5.76 − 5.76i)23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.999 + 0.0154i)5-s + (0.301 + 0.301i)7-s + (0.250 + 0.250i)8-s + (−0.507 + 0.492i)10-s − 0.647i·11-s + (1.09 − 1.09i)13-s − 0.301·14-s − 0.250·16-s + (−0.171 + 0.171i)17-s − 1.08i·19-s + (0.00771 − 0.499i)20-s + (0.323 + 0.323i)22-s + (−1.20 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603580255\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603580255\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.0345i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.796 - 0.796i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.14iT - 11T^{2} \) |
| 13 | \( 1 + (-3.95 + 3.95i)T - 13iT^{2} \) |
| 19 | \( 1 + 4.73iT - 19T^{2} \) |
| 23 | \( 1 + (5.76 + 5.76i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 + (8.37 + 8.37i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.42 - 8.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.461 + 0.461i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + (-2.22 - 2.22i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.51iT - 71T^{2} \) |
| 73 | \( 1 + (3.15 - 3.15i)T - 73iT^{2} \) |
| 79 | \( 1 + 16.2iT - 79T^{2} \) |
| 83 | \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.47 - 1.47i)T + 97iT^{2} \) |
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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
−9.247619931744022697832744523206, −8.504036245101163590113189590618, −8.135058410341679162501801026299, −6.81540888512046828313680671702, −6.16750216338657793664742753590, −5.55452894526234898216147714308, −4.63724939890354605499570099028, −3.18195142561283581577464396852, −2.09851717330866416139491678529, −0.789300300572429488946087989200,
1.46536907558974387935606225513, 1.98391333467159119711145823945, 3.43459326614864230783171692647, 4.32470224629253580424066971661, 5.40977700342048265090353702697, 6.38510325818341766385464467425, 7.07504234469951047087429836703, 8.173128216543983877400166758353, 8.784129165054947557554683321603, 9.711220504121142355629817561833