| L(s) = 1 | + (−3.23 − 2.35i)2-s + (2.78 + 8.55i)3-s + (4.94 + 15.2i)4-s + (−55.1 − 9.31i)5-s + (11.1 − 34.2i)6-s + 58.3·7-s + (19.7 − 60.8i)8-s + (−65.5 + 47.6i)9-s + (156. + 159. i)10-s + (270. + 196. i)11-s + (−116. + 84.6i)12-s + (230. − 167. i)13-s + (−188. − 137. i)14-s + (−73.5 − 497. i)15-s + (−207. + 150. i)16-s + (−147. + 453. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.986 − 0.166i)5-s + (0.126 − 0.388i)6-s + 0.450·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.494 + 0.505i)10-s + (0.673 + 0.489i)11-s + (−0.233 + 0.169i)12-s + (0.378 − 0.274i)13-s + (−0.257 − 0.187i)14-s + (−0.0844 − 0.571i)15-s + (−0.202 + 0.146i)16-s + (−0.123 + 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.002400615813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002400615813\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 5 | \( 1 + (55.1 + 9.31i)T \) |
| good | 7 | \( 1 - 58.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-270. - 196. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-230. + 167. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (147. - 453. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (886. - 2.72e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.63e3 + 1.91e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.56e3 + 7.88e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-3.21e3 + 9.89e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.56e3 - 6.95e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.55e4 - 1.12e4i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 468.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.44e3 + 2.60e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-3.79e3 - 1.16e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.27e4 - 2.37e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.90e4 + 1.38e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-2.58e3 + 7.96e3i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.18e4 + 6.72e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.76e4 - 2.01e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.93e4 - 5.95e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-7.84e3 + 2.41e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-8.72e3 - 6.34e3i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.78e4 - 5.48e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.67303164753674496356926767414, −10.51670994606586823231786133817, −9.697007934432604190878832956147, −8.290381640683013898410796218003, −7.984367818839319059009769740613, −6.27085736606422679015666036899, −4.41619072783347945682621783017, −3.65405676594383270205116320848, −1.81121846894774408688564628226, −0.000979662214225365547114011506,
1.48398759793262070239081645618, 3.34753158594684643035257173310, 4.93836395472753367141752512775, 6.60284093217409738811903965686, 7.27560124998069474827563867553, 8.479006871568421225236592508880, 9.024221478459455042647463468715, 10.76361377141283438485804274568, 11.44357388099737158243009199930, 12.40884977226043984660025021747
