L(s) = 1 | + (0.694 + 3.93i)2-s + (7.96 − 6.68i)3-s + (−15.0 + 5.47i)4-s + (−89.7 − 32.6i)5-s + (31.8 + 26.7i)6-s + (−116. + 201. i)7-s + (−32 − 55.4i)8-s + (−23.4 + 132. i)9-s + (66.3 − 376. i)10-s + (−172. − 299. i)11-s + (−83.2 + 144. i)12-s + (387. + 325. i)13-s + (−872. − 317. i)14-s + (−933. + 339. i)15-s + (196. − 164. i)16-s + (−138. − 784. i)17-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (0.511 − 0.428i)3-s + (−0.469 + 0.171i)4-s + (−1.60 − 0.584i)5-s + (0.361 + 0.303i)6-s + (−0.895 + 1.55i)7-s + (−0.176 − 0.306i)8-s + (−0.0963 + 0.546i)9-s + (0.209 − 1.18i)10-s + (−0.430 − 0.746i)11-s + (−0.166 + 0.288i)12-s + (0.636 + 0.534i)13-s + (−1.19 − 0.433i)14-s + (−1.07 + 0.390i)15-s + (0.191 − 0.160i)16-s + (−0.116 − 0.658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0168227 - 0.390173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0168227 - 0.390173i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 19 | \( 1 + (880. + 1.30e3i)T \) |
good | 3 | \( 1 + (-7.96 + 6.68i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (89.7 + 32.6i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (116. - 201. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (172. + 299. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-387. - 325. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (138. + 784. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-1.11e3 + 407. i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (73.8 - 418. i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (3.32e3 - 5.75e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 7.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.23e4 - 1.03e4i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (1.21e4 + 4.41e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (3.67e3 - 2.08e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-2.73e4 + 9.94e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-1.62e3 - 9.20e3i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-7.81e3 + 2.84e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (2.49e3 - 1.41e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-2.93e4 - 1.06e4i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-1.52e3 + 1.28e3i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (7.37e3 - 6.18e3i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-1.70e4 + 2.96e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (8.52e4 + 7.15e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-1.68e4 - 9.55e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−15.94131995545316852210327273072, −15.03701754966205734859000393333, −13.44893698917723044608219797648, −12.52781023453861168581751673100, −11.37586397748421341443607024638, −8.807724374862911125270534622373, −8.447438249936307293595969163022, −6.90985670202638249715253268611, −5.11373136573686452522482021205, −3.14506258521352934711349685172,
0.19851323887680771422767506815, 3.51296376658486878344653779001, 3.95020936848989029498117147904, 6.94481945451240204236396111111, 8.266369053329343715485251291399, 10.05288165705484501747363169432, 10.79498726298343194051420534487, 12.22329578099164960172265797599, 13.37408385679752733751681831384, 14.83329113562861803582256799446