| L(s) = 1 | + (−0.855 − 0.0448i)2-s + (−0.835 − 1.03i)3-s + (−1.25 − 0.132i)4-s + (1.02 + 1.98i)5-s + (0.668 + 0.920i)6-s + (−0.0574 + 2.64i)7-s + (2.76 + 0.437i)8-s + (0.256 − 1.20i)9-s + (−0.786 − 1.74i)10-s + (0.886 − 0.188i)11-s + (0.915 + 1.41i)12-s + (5.68 + 2.89i)13-s + (0.167 − 2.26i)14-s + (1.19 − 2.71i)15-s + (0.131 + 0.0279i)16-s + (0.276 + 0.720i)17-s + ⋯ |
| L(s) = 1 | + (−0.605 − 0.0317i)2-s + (−0.482 − 0.595i)3-s + (−0.629 − 0.0661i)4-s + (0.457 + 0.889i)5-s + (0.273 + 0.375i)6-s + (−0.0217 + 0.999i)7-s + (0.977 + 0.154i)8-s + (0.0856 − 0.402i)9-s + (−0.248 − 0.552i)10-s + (0.267 − 0.0568i)11-s + (0.264 + 0.407i)12-s + (1.57 + 0.803i)13-s + (0.0448 − 0.604i)14-s + (0.309 − 0.701i)15-s + (0.0328 + 0.00698i)16-s + (0.0671 + 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.682191 + 0.184995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682191 + 0.184995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.02 - 1.98i)T \) |
| 7 | \( 1 + (0.0574 - 2.64i)T \) |
| good | 2 | \( 1 + (0.855 + 0.0448i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.835 + 1.03i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-0.886 + 0.188i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.68 - 2.89i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.276 - 0.720i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.0762 + 0.725i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.328 - 6.25i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (3.24 - 4.46i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.93 + 4.34i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-8.05 + 5.23i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-5.14 + 1.67i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.27 + 3.27i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.361 - 0.138i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (6.15 - 4.98i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (0.380 - 0.422i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-7.85 + 7.07i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.29 - 0.879i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (9.01 + 6.54i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.64 + 10.2i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-0.729 - 1.63i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.805 - 5.08i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-2.15 - 2.38i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-1.57 - 9.95i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.86929668981004339993770949475, −11.57614559842464426416082359412, −10.94771502271461160223110663643, −9.503647333773493249050861031918, −9.033476496813826371687900459408, −7.66025075459226200557310877081, −6.39970906962381934679504879140, −5.67353943700095944025558367807, −3.72674549329707876346902004001, −1.64032126183438281381920837351,
1.01532523295867503228411784738, 4.03209276961467473599327134883, 4.84159331523015928646416953168, 6.12063621472703744175545749736, 7.86969780047023756815930850480, 8.592032961735806861738795616262, 9.801637514091566907673036636788, 10.35944461690107611119240557810, 11.29898391333352342732180572374, 12.99143573156106614197260558710
