L(s) = 1 | + (−1.40 + 0.150i)2-s + (−2.03 + 1.17i)3-s + (1.95 − 0.422i)4-s + (1.50 − 0.869i)5-s + (2.67 − 1.95i)6-s + 2.63i·7-s + (−2.68 + 0.888i)8-s + (1.24 − 2.16i)9-s + (−1.98 + 1.44i)10-s − 3.51·11-s + (−3.47 + 3.15i)12-s + (−3.13 + 5.42i)13-s + (−0.395 − 3.70i)14-s + (−2.03 + 3.53i)15-s + (3.64 − 1.65i)16-s + (−0.535 − 0.926i)17-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.106i)2-s + (−1.17 + 0.676i)3-s + (0.977 − 0.211i)4-s + (0.673 − 0.388i)5-s + (1.09 − 0.797i)6-s + 0.995i·7-s + (−0.949 + 0.314i)8-s + (0.416 − 0.721i)9-s + (−0.628 + 0.458i)10-s − 1.05·11-s + (−1.00 + 0.909i)12-s + (−0.868 + 1.50i)13-s + (−0.105 − 0.989i)14-s + (−0.526 + 0.911i)15-s + (0.910 − 0.413i)16-s + (−0.129 − 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159488 + 0.358766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159488 + 0.358766i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.150i)T \) |
| 19 | \( 1 + (3.79 - 2.13i)T \) |
good | 3 | \( 1 + (2.03 - 1.17i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.50 + 0.869i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2.63iT - 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 + (3.13 - 5.42i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.535 + 0.926i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.79 - 3.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.579 + 1.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 + (4.54 - 2.62i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.164 + 0.284i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.68 + 3.85i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.526 + 0.912i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.45 + 4.87i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.2 - 6.51i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 - 7.14i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.554 - 0.960i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.98 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.02 + 1.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + (-2.18 - 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.354 + 0.204i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.07198150447017181009542834680, −11.86957548549253117156697733568, −11.31790622224040119048737684627, −10.12079898905000657518981930760, −9.510069578799789947484419964091, −8.462453323928998850454228331966, −6.86160849824854706080528969865, −5.73428634657069796128603259012, −4.95605063204108376093152880730, −2.23742134881219737919946836810,
0.56959409656755576282069745426, 2.62462147807573608515204523924, 5.24273398533689552199771973070, 6.44616294246887925247660694609, 7.21052642814610161992638847233, 8.259482233621279269709074831408, 10.05846350081633618605853825486, 10.50547801466043374323017483208, 11.27551056673925487531886988860, 12.66593257911755738900846260653