L(s) = 1 | + (0.694 − 3.93i)2-s + (−2.94 − 16.7i)3-s + (−15.0 − 5.47i)4-s + (46.4 − 38.9i)5-s − 67.8·6-s + (193. − 162. i)7-s + (−32 + 55.4i)8-s + (−42.0 + 15.3i)9-s + (−121. − 209. i)10-s + (36.9 − 63.9i)11-s + (−47.1 + 267. i)12-s + (−41.1 − 14.9i)13-s + (−505. − 875. i)14-s + (−787. − 660. i)15-s + (196. + 164. i)16-s + (−893. + 325. i)17-s + ⋯ |
L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.188 − 1.07i)3-s + (−0.469 − 0.171i)4-s + (0.830 − 0.696i)5-s − 0.769·6-s + (1.49 − 1.25i)7-s + (−0.176 + 0.306i)8-s + (−0.173 + 0.0629i)9-s + (−0.383 − 0.663i)10-s + (0.0920 − 0.159i)11-s + (−0.0944 + 0.535i)12-s + (−0.0674 − 0.0245i)13-s + (−0.688 − 1.19i)14-s + (−0.903 − 0.758i)15-s + (0.191 + 0.160i)16-s + (−0.750 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.328620 - 2.18290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.328620 - 2.18290i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.694 + 3.93i)T \) |
| 37 | \( 1 + (-8.17e3 - 1.56e3i)T \) |
good | 3 | \( 1 + (2.94 + 16.7i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (-46.4 + 38.9i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-193. + 162. i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-36.9 + 63.9i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (41.1 + 14.9i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (893. - 325. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 19 | \( 1 + (-367. - 2.08e3i)T + (-2.32e6 + 8.46e5i)T^{2} \) |
| 23 | \( 1 + (-1.46e3 - 2.52e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-196. + 340. i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 529.T + 2.86e7T^{2} \) |
| 41 | \( 1 + (7.21e3 + 2.62e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-9.83e3 - 1.70e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.11e4 - 9.36e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-2.33e4 - 1.96e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-2.32e4 - 8.45e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (1.58e4 - 1.33e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (320. + 1.81e3i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + 5.81e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-6.08e4 + 5.10e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (8.26e4 - 3.00e4i)T + (3.01e9 - 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-3.70e4 - 3.10e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (2.82e4 + 4.89e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.28181862656914746609035297580, −12.03980272234866491025760812733, −11.06337189338513237607942965626, −9.887428569327268140836444129660, −8.377623585862147762169724057827, −7.28069132096380960997174267675, −5.61515379188715250617482010363, −4.25513192459284611124903743021, −1.71050898738368253047527268986, −1.11471158994031501971529439984,
2.39009003343601583389562050957, 4.62850984300363924161964751344, 5.38410275970719987596726004960, 6.82261750050315581223934045554, 8.522182264150530467879843718950, 9.443200560166624999503349198659, 10.69859999947399404413929048203, 11.64575496020106108498181642347, 13.30351219048590228848639696070, 14.59166627509289879765701063912