L(s) = 1 | + (−0.258 + 0.965i)2-s + (3.03 + 0.814i)3-s + (−0.866 − 0.499i)4-s + (−1.57 + 2.72i)6-s + (−2.05 + 2.05i)7-s + (0.707 − 0.707i)8-s + (5.96 + 3.44i)9-s − 5.34·11-s + (−2.22 − 2.22i)12-s + (1.00 + 3.75i)13-s + (−1.45 − 2.51i)14-s + (0.500 + 0.866i)16-s + (2.56 + 0.687i)17-s + (−4.87 + 4.87i)18-s + (−3.18 + 2.97i)19-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (1.75 + 0.469i)3-s + (−0.433 − 0.249i)4-s + (−0.641 + 1.11i)6-s + (−0.777 + 0.777i)7-s + (0.249 − 0.249i)8-s + (1.98 + 1.14i)9-s − 1.61·11-s + (−0.641 − 0.641i)12-s + (0.279 + 1.04i)13-s + (−0.388 − 0.673i)14-s + (0.125 + 0.216i)16-s + (0.622 + 0.166i)17-s + (−1.14 + 1.14i)18-s + (−0.730 + 0.682i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.791857 + 1.90893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.791857 + 1.90893i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.18 - 2.97i)T \) |
good | 3 | \( 1 + (-3.03 - 0.814i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (2.05 - 2.05i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 + (-1.00 - 3.75i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 0.687i)T + (14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (-2.80 + 0.752i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.53 + 6.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.91iT - 31T^{2} \) |
| 37 | \( 1 + (3.67 + 3.67i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.61 + 2.66i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 6.51i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-0.622 - 2.32i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.917 - 3.42i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.58 - 7.93i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.199 + 0.345i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0355 + 0.00951i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.91 + 2.26i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.16 + 15.5i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 1.99i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.45 - 3.45i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.68 + 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.59 + 13.4i)T + (-84.0 - 48.5i)T^{2} \) |
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
−10.14956799526280951181058194346, −9.143630351798926660308173129975, −8.779741321836720028041901604720, −7.996315854973383227470159635436, −7.27378634236939640447011987342, −6.16301886768548969306818075133, −5.06480493096350199102006258243, −4.00183948721249909173362701913, −3.00187145986531243043824249515, −2.11654436493519132353888765115,
0.828074666296963474884691264127, 2.43963723788457692717770721810, 3.03345996468168181883033302677, 3.80371402648376839451016200312, 5.09658662962307474094993605717, 6.66087954303855204475756326185, 7.62306061343080290171423555875, 8.055791161353791884957602381909, 8.867338012690981962640587302673, 9.830088341046468986067352766432