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 | \( 1 + (10.5 - 6.08i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.13810026655952637233270399102, −14.12525767538565813791760632012, −13.33103850557207692695489901720, −11.33012638308623222711387126149, −10.77143993004578887043044939915, −9.704895302553094668753371638454, −7.84857580399543244253049700742, −6.08625732330117526726608202729, −4.99175200570362950331213995770, −4.03098402667033651751282679777,
2.21990749613258928673505807673, 4.76562561577916403878006924316, 5.73154363345460667563288823068, 7.68422755443374245791796019238, 8.456886944885747281753485335411, 10.75958084177328027005701448713, 11.76531640754269803092495395836, 12.76740594947765634018058348632, 12.97207392389289149231656221357, 14.42302680234317028274053547927