| L(s) = 1 | + (1.23 + 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (31.6 − 46.0i)5-s + (29.1 + 21.1i)6-s + 113.·7-s + (−51.7 − 37.6i)8-s + (25.0 − 77.0i)9-s + (214. + 63.6i)10-s + (−165. − 507. i)11-s + (−44.4 + 136. i)12-s + (−89.1 + 274. i)13-s + (140. + 433. i)14-s + (−12.9 − 502. i)15-s + (79.1 − 243. i)16-s + (−1.18e3 − 863. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.566 − 0.823i)5-s + (0.330 + 0.239i)6-s + 0.878·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (0.677 + 0.201i)10-s + (−0.411 − 1.26i)11-s + (−0.0892 + 0.274i)12-s + (−0.146 + 0.450i)13-s + (0.191 + 0.590i)14-s + (−0.0148 − 0.577i)15-s + (0.0772 − 0.237i)16-s + (−0.997 − 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.591686997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591686997\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-31.6 + 46.0i)T \) |
| good | 7 | \( 1 - 113.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (165. + 507. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (89.1 - 274. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.18e3 + 863. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-381. - 277. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (366. + 1.12e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.17e3 + 3.03e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-621. - 451. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.31e3 + 1.01e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (5.02e3 - 1.54e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (605. - 439. i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.05e4 + 1.49e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.02e3 + 9.29e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-8.60e3 - 2.64e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (8.23e3 + 5.98e3i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-5.85e4 + 4.25e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.18e4 - 3.64e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.78e3 + 1.29e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.62e3 - 1.18e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-1.55e4 - 4.78e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.59e4 - 4.06e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.19253689583991949370532700669, −11.09202422000469090527854805933, −9.529310228471040693790807656739, −8.591230341335637636321531842334, −7.953288928199883429287629200422, −6.53160664707729219653883386056, −5.39048739870341335845058464654, −4.31989188291102943867420944368, −2.43717370788257511194258401132, −0.77569958670427977000343407188,
1.76426094347359706001412888934, 2.72590786403866640251958092454, 4.25091635323234037185864847373, 5.35186536271386867356791697335, 6.95398215277178995945998229996, 8.181828666289680772125050381734, 9.444049856734935638529161197842, 10.35875659649461304976468089692, 10.97888135205292600223329472543, 12.22943113497906897209900900171
