| 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 | \( 1 + (2.82e4 - 4.89e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.59166627509289879765701063912, −13.30351219048590228848639696070, −11.64575496020106108498181642347, −10.69859999947399404413929048203, −9.443200560166624999503349198659, −8.522182264150530467879843718950, −6.82261750050315581223934045554, −5.38410275970719987596726004960, −4.62850984300363924161964751344, −2.39009003343601583389562050957,
1.11471158994031501971529439984, 1.71050898738368253047527268986, 4.25513192459284611124903743021, 5.61515379188715250617482010363, 7.28069132096380960997174267675, 8.377623585862147762169724057827, 9.887428569327268140836444129660, 11.06337189338513237607942965626, 12.03980272234866491025760812733, 13.28181862656914746609035297580
