L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.866 − 0.5i)4-s + (0.965 + 0.258i)7-s + 1.00i·9-s + (−1.5 + 0.866i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (0.499 + 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)28-s + (1.67 + 0.448i)33-s + (0.500 − 0.866i)36-s + (−0.866 − 0.500i)39-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.866 − 0.5i)4-s + (0.965 + 0.258i)7-s + 1.00i·9-s + (−1.5 + 0.866i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (0.499 + 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)28-s + (1.67 + 0.448i)33-s + (0.500 − 0.866i)36-s + (−0.866 − 0.500i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7563486057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7563486057\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
good | 2 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.802205178814420101980683834193, −8.433304791351842122232319872898, −8.158670414228431871936573126649, −7.33796666494495767414590105028, −6.09862399986143329837622314461, −5.41561688202049609837028042387, −4.98803218512581441586321275408, −3.85822888133330841843386046942, −2.18651234074069398705553299977, −1.18192941330395818972174833013,
0.846054147690045600057702797730, 2.97688283374728085744946705221, 3.78803479193688569225931744245, 4.84607941391434299780360790753, 5.23834502962344823948860588273, 6.10133970460216124444079793611, 7.54139980727082895643937960653, 8.026197242239290354142187254694, 8.897807471904525950032913410162, 9.603038851836469060998516007530