L(s) = 1 | + (0.965 − 0.258i)3-s + (0.258 − 0.965i)5-s + (0.5 − 0.866i)11-s + (−0.707 − 0.707i)13-s − i·15-s + (−0.258 − 0.965i)17-s + (−1.22 + 0.707i)19-s + (−0.866 − 0.499i)25-s + (−0.707 + 0.707i)27-s − i·29-s + (0.707 − 1.22i)31-s + (0.258 − 0.965i)33-s + (−0.866 − 0.500i)39-s − 1.41·41-s + (1 + i)43-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)3-s + (0.258 − 0.965i)5-s + (0.5 − 0.866i)11-s + (−0.707 − 0.707i)13-s − i·15-s + (−0.258 − 0.965i)17-s + (−1.22 + 0.707i)19-s + (−0.866 − 0.499i)25-s + (−0.707 + 0.707i)27-s − i·29-s + (0.707 − 1.22i)31-s + (0.258 − 0.965i)33-s + (−0.866 − 0.500i)39-s − 1.41·41-s + (1 + i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.626205277\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626205277\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
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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
−8.531979959843960025971898345526, −7.929314483470539353951250453683, −7.20427602444384856416951772946, −6.06190182280122555265186221582, −5.58055084041084719346600847061, −4.54482047876721031802515851187, −3.83279218689555861063668990937, −2.71314634829210451204605601684, −2.13447574703808811893534114010, −0.76648817843108216471790171746,
1.94558355682392637657756211720, 2.39377036360376135538060140204, 3.43060333101481524349357445949, 4.08225916803828420460199462043, 4.93872318481147057708671776408, 6.09669797041824594924444694630, 6.86475181942245792756568562871, 7.17820773697634388504744802451, 8.304971174467234120744749281895, 8.885351695668843587362374969566