| L(s) = 1 | + (−0.295 − 1.38i)2-s + (0.471 + 0.881i)3-s + (−1.82 + 0.816i)4-s + (−0.528 + 0.433i)5-s + (1.08 − 0.912i)6-s + (1.34 − 0.901i)7-s + (1.66 + 2.28i)8-s + (−0.555 + 0.831i)9-s + (0.756 + 0.603i)10-s + (1.63 + 5.38i)11-s + (−1.58 − 1.22i)12-s + (−2.08 + 2.53i)13-s + (−1.64 − 1.60i)14-s + (−0.631 − 0.261i)15-s + (2.66 − 2.98i)16-s + (3.13 − 1.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.208 − 0.977i)2-s + (0.272 + 0.509i)3-s + (−0.912 + 0.408i)4-s + (−0.236 + 0.194i)5-s + (0.441 − 0.372i)6-s + (0.510 − 0.340i)7-s + (0.589 + 0.807i)8-s + (−0.185 + 0.277i)9-s + (0.239 + 0.190i)10-s + (0.492 + 1.62i)11-s + (−0.456 − 0.353i)12-s + (−0.577 + 0.703i)13-s + (−0.439 − 0.427i)14-s + (−0.163 − 0.0675i)15-s + (0.666 − 0.745i)16-s + (0.761 − 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18418 + 0.145231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18418 + 0.145231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.295 + 1.38i)T \) |
| 3 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 5 | \( 1 + (0.528 - 0.433i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 0.901i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 5.38i)T + (-9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (2.08 - 2.53i)T + (-2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 1.30i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (4.93 - 0.486i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-9.20 + 1.83i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-8.34 - 2.53i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (1.13 + 1.13i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.101 + 1.03i)T + (-36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.931 - 4.68i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.71 + 5.08i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.59 + 3.84i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (9.08 - 2.75i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (7.18 + 8.75i)T + (-11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (4.05 - 2.16i)T + (33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-5.25 + 2.80i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (5.21 + 7.81i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (7.06 + 4.71i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.16 + 5.21i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.45 - 14.7i)T + (-81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (5.09 + 1.01i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (2.29 + 2.29i)T + 97iT^{2} \) |
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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
−11.21783947776870977591076119021, −10.51056721387249576206010243140, −9.609073597696661984142594719996, −8.984411401971741215593725987245, −7.79765687954508692925400848790, −6.92959550916293130616170155065, −4.83777572637955441981193529735, −4.43703373067947873492415504257, −3.07719325032525549142357494446, −1.71619987739610869971266690533,
0.935607551173798866385190270625, 3.11791770729892343665825242798, 4.62039122690101719675335587107, 5.73477505551260094669060535126, 6.54765390465306349412810596366, 7.73682798253598528760669155344, 8.426822041499951871668436313846, 8.963157799927384014530529685126, 10.27770602843753783534704518862, 11.27882513771396347990154526484
