L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.317 + 2.20i)3-s + (0.415 − 0.909i)4-s + (1.95 − 0.573i)5-s + (−0.925 − 2.02i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−1.88 − 0.554i)9-s + (−1.33 + 1.53i)10-s + (3.96 + 2.55i)11-s + (1.87 + 1.20i)12-s + (0.898 − 1.03i)13-s + (−0.959 − 0.281i)14-s + (0.645 + 4.48i)15-s + (−0.654 − 0.755i)16-s + (−0.290 − 0.637i)17-s + ⋯ |
L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.183 + 1.27i)3-s + (0.207 − 0.454i)4-s + (0.872 − 0.256i)5-s + (−0.377 − 0.827i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.629 − 0.184i)9-s + (−0.421 + 0.486i)10-s + (1.19 + 0.768i)11-s + (0.541 + 0.347i)12-s + (0.249 − 0.287i)13-s + (−0.256 − 0.0752i)14-s + (0.166 + 1.15i)15-s + (−0.163 − 0.188i)16-s + (−0.0705 − 0.154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.721114 + 0.889504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.721114 + 0.889504i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.71 + 0.856i)T \) |
good | 3 | \( 1 + (0.317 - 2.20i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-1.95 + 0.573i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-3.96 - 2.55i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.898 + 1.03i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.290 + 0.637i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.0440 + 0.0965i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.692 + 1.51i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.731 - 5.08i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (8.95 + 2.62i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-6.87 + 2.01i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0826 + 0.574i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + (3.27 + 3.77i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-9.39 + 10.8i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.249 - 1.73i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.16 + 2.67i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-8.96 + 5.76i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.72 - 5.97i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (10.4 - 12.0i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.15 - 0.631i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.90 + 13.2i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-15.1 + 4.44i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.61501996468670351211356998496, −10.61949718910198838473755579143, −9.810197106786137532230431096140, −9.335964577323261410101700087383, −8.456739533410668673418951638459, −7.01500159688385077322025347034, −5.88587784698603937568345753673, −5.02049645747821383236449716824, −3.87898722201574075842081127582, −1.84806010897408607257567588641,
1.17859864469636960180561333639, 2.19424205659108371975415240503, 3.89203799730654321833026863603, 5.92492967271778334862318426422, 6.53558388512873852092821823182, 7.52144092514377142278340373735, 8.521100645310652294653988416118, 9.483417904394980849842021458377, 10.46818340336656222347277930262, 11.52076778115850210160476991973