L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.499i)6-s + (−1.73 + i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.401 + 0.232i)11-s − 0.999·12-s + (−1 + 3.46i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + 0.999i·18-s + (−0.464 + 0.267i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.654 + 0.377i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.121 + 0.0699i)11-s − 0.288·12-s + (−0.277 + 0.960i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + 0.235i·18-s + (−0.106 + 0.0614i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3718411986\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3718411986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.401 - 0.232i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.464 - 0.267i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.133 + 0.232i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.86 - 3.23i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (1.03 + 0.598i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.73 + i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.964 + 1.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + (1.33 - 0.767i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.19 + 9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.92 + 2.26i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.26 - 4.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 0.0717T + 79T^{2} \) |
| 83 | \( 1 + 4.92iT - 83T^{2} \) |
| 89 | \( 1 + (-6.46 - 3.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.46 + 3.73i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.107778937735019130027753548622, −8.531314944205767214496533073530, −7.27385877679319593281333610170, −6.75693075711420627702303961302, −5.80366352473998086450495290098, −4.80841350826342263292781785553, −3.89629049687294735001219729209, −2.92581257043853707049120219946, −1.89127148038880846063393944289, −0.19451709011601680327916292132,
1.05982964508909167620417089179, 2.36149326721524111922882274802, 3.48973585444318246460146564919, 4.67666284124574887771239654372, 5.77397555184264790586563006061, 6.30288734498247466651051855860, 7.13870872803815850785751799767, 7.78123059565408424367090898443, 8.549473870246104668795070356915, 9.337294403162206471632822852734