L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.41 − 2.44i)3-s + (−0.499 − 0.866i)4-s + 2.82·6-s + 0.999·8-s + (−2.49 + 4.33i)9-s + (0.5 + 0.866i)11-s + (−1.41 + 2.44i)12-s + 4.24·13-s + (−0.5 + 0.866i)16-s + (1.41 + 2.44i)17-s + (−2.5 − 4.33i)18-s + (2.12 − 3.67i)19-s − 0.999·22-s + (−3 + 5.19i)23-s + (−1.41 − 2.44i)24-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.816 − 1.41i)3-s + (−0.249 − 0.433i)4-s + 1.15·6-s + 0.353·8-s + (−0.833 + 1.44i)9-s + (0.150 + 0.261i)11-s + (−0.408 + 0.707i)12-s + 1.17·13-s + (−0.125 + 0.216i)16-s + (0.342 + 0.594i)17-s + (−0.589 − 1.02i)18-s + (0.486 − 0.842i)19-s − 0.213·22-s + (−0.625 + 1.08i)23-s + (−0.288 − 0.499i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9919968701\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9919968701\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 3.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-3.53 - 6.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-6.36 + 11.0i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.65 + 9.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.94 + 8.57i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-4.24 - 7.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−9.714211581038556159289075838540, −8.784229410582820689897826088916, −7.940360557518477234317406468495, −7.23578761412582433242552160222, −6.58264945385010845091290977510, −5.81593142170049109768435747105, −5.13649040754469950502269705972, −3.60119681789291084263510779006, −1.85187184812690158034404391217, −0.905456057023445651646997909672,
0.835246760862546430766975005413, 2.74174831006288833916842200838, 3.93916196950605438771150927825, 4.36382061921744176456898733077, 5.65331695370207060271284626456, 6.16565291277193775863964397411, 7.64052900021886298656851426515, 8.625616295952235757414077206119, 9.356936636813252356059572395977, 10.04782721068340521854276309270