| L(s) = 1 | + (1.23 − 3.80i)2-s + (−7.28 − 5.29i)3-s + (−12.9 − 9.40i)4-s + (−55.5 − 5.99i)5-s + (−29.1 + 21.1i)6-s + 161.·7-s + (−51.7 + 37.6i)8-s + (25.0 + 77.0i)9-s + (−91.5 + 204. i)10-s + (−141. + 435. i)11-s + (44.4 + 136. i)12-s + (78.8 + 242. i)13-s + (199. − 612. i)14-s + (372. + 337. i)15-s + (79.1 + 243. i)16-s + (−1.33e3 + 969. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.994 − 0.107i)5-s + (−0.330 + 0.239i)6-s + 1.24·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.289 + 0.645i)10-s + (−0.352 + 1.08i)11-s + (0.0892 + 0.274i)12-s + (0.129 + 0.398i)13-s + (0.271 − 0.835i)14-s + (0.427 + 0.387i)15-s + (0.0772 + 0.237i)16-s + (−1.11 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.489503912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489503912\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 3 | \( 1 + (7.28 + 5.29i)T \) |
| 5 | \( 1 + (55.5 + 5.99i)T \) |
| good | 7 | \( 1 - 161.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (141. - 435. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-78.8 - 242. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.33e3 - 969. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.28e3 + 930. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-1.03e3 + 3.18e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.15e3 - 3.01e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-8.15e3 + 5.92e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (562. + 1.73e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-577. - 1.77e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.10e4 - 1.52e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.91e4 - 1.38e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.89e3 - 5.84e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.66e3 - 5.13e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-2.49e4 + 1.81e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.33e4 + 9.67e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (970. - 2.98e3i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.83e4 - 3.51e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.72e4 + 1.97e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (4.02e4 - 1.23e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (4.59e4 + 3.33e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−11.92199769023139419796169076092, −11.21077929196258832941475837800, −10.40750399057160877981650276907, −8.802399401973759585131268612003, −7.86265144776138774957217070820, −6.69446157609143919674499863463, −4.86513480769833160296108157667, −4.36896031414309125584397743801, −2.37079076684335310477010183364, −0.951042670139335569548715681878,
0.70131209110211557782226002304, 3.28071768056791266733495734025, 4.62214417380094266687266176709, 5.43598401225983654365548982486, 6.89952584984878136052424080558, 8.008754061873173434561498800880, 8.683731896101203706012020399884, 10.34271388090650016593159785356, 11.48918639178241442217553027719, 11.81677514589242313458063877965
