L(s) = 1 | + (−1.5 + 2.59i)5-s − 2.64·7-s + (−1.32 − 2.29i)11-s − 4·13-s + (0.5 + 0.866i)17-s + (3.96 − 6.87i)19-s + (−1.32 + 2.29i)23-s + (−2 − 3.46i)25-s + 4·29-s + (1.32 + 2.29i)31-s + (3.96 − 6.87i)35-s + (2.5 − 4.33i)37-s − 8·41-s + 10.5·43-s + (1.32 − 2.29i)47-s + ⋯ |
L(s) = 1 | + (−0.670 + 1.16i)5-s − 0.999·7-s + (−0.398 − 0.690i)11-s − 1.10·13-s + (0.121 + 0.210i)17-s + (0.910 − 1.57i)19-s + (−0.275 + 0.477i)23-s + (−0.400 − 0.692i)25-s + 0.742·29-s + (0.237 + 0.411i)31-s + (0.670 − 1.16i)35-s + (0.410 − 0.711i)37-s − 1.24·41-s + 1.61·43-s + (0.192 − 0.334i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9250146640\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9250146640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.32 + 2.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.96 + 6.87i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.32 - 2.29i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-1.32 - 2.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-1.32 + 2.29i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.5 - 6.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.32 - 2.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.32 + 2.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 2.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.221495686680098656871135458962, −8.213858699906801333668902396024, −7.20570839079652354057563568316, −7.05000896865405432530097841898, −6.02336448158187654924155466510, −5.11830150325297986493532674103, −3.96186882139899164203014525417, −3.02439394334540451085005121174, −2.62785971823998841128579185016, −0.46202895887234430552831230870,
0.828907793163875102638107506399, 2.33063206810995252516373757938, 3.45311620282571182134148518719, 4.38412475831685560382062743471, 5.07376860927659775474446478432, 5.95108204420574895342606301619, 6.99465161050351517657755279714, 7.74802488653439631891245282891, 8.321482675951297588541211606443, 9.318944245596956087531850596608