| L(s) = 1 | + (−0.907 − 2.79i)2-s + (−2.42 + 1.76i)3-s + (−0.510 + 0.370i)4-s + (10.7 + 3.16i)5-s + (7.13 + 5.18i)6-s + 18.9·7-s + (−17.5 − 12.7i)8-s + (2.78 − 8.55i)9-s + (−0.895 − 32.8i)10-s + (−1.87 − 5.77i)11-s + (0.584 − 1.79i)12-s + (24.1 − 74.4i)13-s + (−17.1 − 52.8i)14-s + (−31.6 + 11.2i)15-s + (−21.2 + 65.2i)16-s + (−31.0 − 22.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.320 − 0.987i)2-s + (−0.467 + 0.339i)3-s + (−0.0637 + 0.0463i)4-s + (0.959 + 0.282i)5-s + (0.485 + 0.352i)6-s + 1.02·7-s + (−0.774 − 0.562i)8-s + (0.103 − 0.317i)9-s + (−0.0283 − 1.03i)10-s + (−0.0513 − 0.158i)11-s + (0.0140 − 0.0432i)12-s + (0.515 − 1.58i)13-s + (−0.327 − 1.00i)14-s + (−0.544 + 0.193i)15-s + (−0.331 + 1.02i)16-s + (−0.443 − 0.322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.06167 - 0.943837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06167 - 0.943837i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 - 1.76i)T \) |
| 5 | \( 1 + (-10.7 - 3.16i)T \) |
| good | 2 | \( 1 + (0.907 + 2.79i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 + (1.87 + 5.77i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-24.1 + 74.4i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (31.0 + 22.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-75.2 - 54.6i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (25.0 + 77.0i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-9.02 + 6.55i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-181. - 131. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (33.2 - 102. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (108. - 333. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-236. + 171. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (554. - 402. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (69.2 - 213. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (215. + 663. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-735. - 534. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (163. - 118. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (90.2 + 277. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (573. - 416. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-897. - 651. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-1.16 - 3.59i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (14.9 - 10.8i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.65137125943398858515369627965, −12.39397973774624491128290521549, −11.28903250063180151110813673764, −10.51539689651732496552305266991, −9.788149141630983937994296434685, −8.306935203620432257649412238613, −6.36532940662397212418029658962, −5.17670510613684921939326904652, −3.00638107745868458453325005419, −1.26323475153528090342522070661,
1.85960691261298093784710488559, 4.90563138169284900731942802917, 6.11356365135798325769511737424, 7.11098789321572505504910424030, 8.407810933919770019114626858827, 9.447564149767908114750422095133, 11.16591588010285029951374901966, 11.95911695906000600469855483756, 13.54191524913567097483990714742, 14.27094271103581721183143838967
