L(s) = 1 | + (1.05 − 0.609i)2-s + (−1.16 − 2.01i)3-s + (−0.256 + 0.443i)4-s − i·5-s + (−2.46 − 1.42i)6-s + (3.11 + 1.80i)7-s + 3.06i·8-s + (−1.21 + 2.11i)9-s + (−0.609 − 1.05i)10-s + (−4.65 + 2.68i)11-s + 1.19·12-s + (1.81 − 3.11i)13-s + 4.39·14-s + (−2.01 + 1.16i)15-s + (1.35 + 2.34i)16-s + (−0.565 + 0.980i)17-s + ⋯ |
L(s) = 1 | + (0.746 − 0.431i)2-s + (−0.673 − 1.16i)3-s + (−0.128 + 0.221i)4-s − 0.447i·5-s + (−1.00 − 0.580i)6-s + (1.17 + 0.680i)7-s + 1.08i·8-s + (−0.406 + 0.704i)9-s + (−0.192 − 0.334i)10-s + (−1.40 + 0.809i)11-s + 0.344·12-s + (0.504 − 0.863i)13-s + 1.17·14-s + (−0.521 + 0.301i)15-s + (0.339 + 0.587i)16-s + (−0.137 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.895026 - 0.527727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.895026 - 0.527727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 + (-1.81 + 3.11i)T \) |
good | 2 | \( 1 + (-1.05 + 0.609i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.11 - 1.80i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.65 - 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.565 - 0.980i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 + 3.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0123 - 0.0214i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (-7.53 + 4.35i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.565 - 0.980i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.58iT - 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + (0.148 + 0.0857i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.54 + 3.19i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.35 - 5.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.5 + 6.08i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.42302680234317028274053547927, −12.97207392389289149231656221357, −12.76740594947765634018058348632, −11.76531640754269803092495395836, −10.75958084177328027005701448713, −8.456886944885747281753485335411, −7.68422755443374245791796019238, −5.73154363345460667563288823068, −4.76562561577916403878006924316, −2.21990749613258928673505807673,
4.03098402667033651751282679777, 4.99175200570362950331213995770, 6.08625732330117526726608202729, 7.84857580399543244253049700742, 9.704895302553094668753371638454, 10.77143993004578887043044939915, 11.33012638308623222711387126149, 13.33103850557207692695489901720, 14.12525767538565813791760632012, 15.13810026655952637233270399102