| L(s) = 1 | + (−6.91 − 9.51i)2-s + (−4.81 − 14.8i)3-s + (−22.9 + 70.5i)4-s + (174. + 127. i)5-s + (−107. + 148. i)6-s + (472. + 153. i)7-s + (114. − 37.0i)8-s + (−196. + 142. i)9-s − 2.54e3i·10-s + (360. + 1.28e3i)11-s + 1.15e3·12-s + (−760. − 1.04e3i)13-s + (−1.80e3 − 5.55e3i)14-s + (1.04e3 − 3.20e3i)15-s + (2.70e3 + 1.96e3i)16-s + (−1.36e3 + 1.87e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.863 − 1.18i)2-s + (−0.178 − 0.549i)3-s + (−0.358 + 1.10i)4-s + (1.39 + 1.01i)5-s + (−0.498 + 0.686i)6-s + (1.37 + 0.447i)7-s + (0.222 − 0.0724i)8-s + (−0.269 + 0.195i)9-s − 2.54i·10-s + (0.270 + 0.962i)11-s + 0.669·12-s + (−0.346 − 0.476i)13-s + (−0.657 − 2.02i)14-s + (0.308 − 0.949i)15-s + (0.659 + 0.479i)16-s + (−0.277 + 0.381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.21697 - 0.597810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21697 - 0.597810i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.81 + 14.8i)T \) |
| 11 | \( 1 + (-360. - 1.28e3i)T \) |
| good | 2 | \( 1 + (6.91 + 9.51i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (-174. - 127. i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-472. - 153. i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (760. + 1.04e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (1.36e3 - 1.87e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-1.11e4 + 3.60e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 + 1.11e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-3.16e3 - 1.02e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (1.45e4 - 1.05e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-2.17e4 + 6.68e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-4.14e4 + 1.34e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 7.93e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.48e4 - 4.56e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-7.63e3 + 5.54e3i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (8.71e4 - 2.68e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-1.01e5 + 1.39e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 + 4.60e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (2.05e5 + 1.49e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-3.17e5 - 1.03e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-6.21e4 - 8.55e4i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (2.02e5 - 2.79e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 2.04e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.16e5 + 8.47e4i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−14.91276488776432980072042001089, −13.95486281126836472928963246217, −12.38296295009571273594346484082, −11.28650373976796567307659308852, −10.30100251827555631901042931715, −9.194451895795212540382690483381, −7.48449897343884879400298516352, −5.62920624027166749284117878036, −2.46970914173553541759556242041, −1.58471016730826228260126516968,
1.15329297746665354408044058963, 4.96189588375575935549272709535, 6.01328307890348792801065736310, 7.919404715370422133453935331842, 9.052975360212669604637836366001, 9.946569996704182753710819554620, 11.67639243919809437319022141165, 13.76678974703391269503821606594, 14.47941570235738570430920825229, 16.16837396874080076818342005519
