L(s) = 1 | + (1.94 − 0.470i)2-s + (1.21 − 2.74i)3-s + (3.55 − 1.82i)4-s + (−5.15 + 2.97i)5-s + (1.07 − 5.90i)6-s + (4.09 + 2.36i)7-s + (6.05 − 5.23i)8-s + (−6.04 − 6.66i)9-s + (−8.62 + 8.21i)10-s + (−6.94 + 12.0i)11-s + (−0.695 − 11.9i)12-s + (−4.03 + 2.32i)13-s + (9.06 + 2.66i)14-s + (1.89 + 17.7i)15-s + (9.30 − 13.0i)16-s + 21.5·17-s + ⋯ |
L(s) = 1 | + (0.971 − 0.235i)2-s + (0.405 − 0.914i)3-s + (0.889 − 0.457i)4-s + (−1.03 + 0.595i)5-s + (0.178 − 0.983i)6-s + (0.584 + 0.337i)7-s + (0.756 − 0.653i)8-s + (−0.671 − 0.740i)9-s + (−0.862 + 0.821i)10-s + (−0.631 + 1.09i)11-s + (−0.0579 − 0.998i)12-s + (−0.310 + 0.179i)13-s + (0.647 + 0.190i)14-s + (0.126 + 1.18i)15-s + (0.581 − 0.813i)16-s + 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.89962 - 0.825289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89962 - 0.825289i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.94 + 0.470i)T \) |
| 3 | \( 1 + (-1.21 + 2.74i)T \) |
good | 5 | \( 1 + (5.15 - 2.97i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.09 - 2.36i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (6.94 - 12.0i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.03 - 2.32i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 21.5T + 289T^{2} \) |
| 19 | \( 1 - 3.83T + 361T^{2} \) |
| 23 | \( 1 + (30.0 - 17.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (39.3 + 22.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-31.8 + 18.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 36.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (10.2 + 17.7i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.50 + 6.07i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-53.1 - 30.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 58.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-11.0 - 19.1i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-47.1 - 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.9 - 98.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 8.30iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 114.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (47.5 + 27.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-15.6 + 27.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 47.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (28.7 - 49.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.35153342073536138336541809794, −13.10469859168917960056152230980, −11.99281451022278166990364112211, −11.58609938057643879727044951567, −9.971142854177314013586917677311, −7.78951805758230313497049358420, −7.34179912687100773735319211787, −5.64470446746348709837693801236, −3.83777587102791590032956904223, −2.23450776955723886053912749451,
3.26764565339518222936703924868, 4.44522909869115440012299941426, 5.54320899621954174830605810138, 7.78797625395337406606928107238, 8.343704407665327638083638063631, 10.33308015551067925665768982268, 11.37165234153289582890000232356, 12.34032018034109356769123450590, 13.75443078766379290950950703816, 14.51942177539976786743870185610