L(s) = 1 | + (0.100 − 0.310i)2-s + (−1.38 − 1.00i)3-s + (1.53 + 1.11i)4-s + (−0.453 + 0.329i)6-s − 3.42·7-s + (1.02 − 0.747i)8-s + (−0.0177 − 0.0546i)9-s + (−1.65 + 5.07i)11-s + (−1.00 − 3.08i)12-s + (1.08 + 3.34i)13-s + (−0.345 + 1.06i)14-s + (1.04 + 3.20i)16-s + (2.06 − 1.50i)17-s − 0.0187·18-s + (1.63 − 1.19i)19-s + ⋯ |
L(s) = 1 | + (0.0713 − 0.219i)2-s + (−0.801 − 0.582i)3-s + (0.765 + 0.556i)4-s + (−0.185 + 0.134i)6-s − 1.29·7-s + (0.363 − 0.264i)8-s + (−0.00592 − 0.0182i)9-s + (−0.497 + 1.53i)11-s + (−0.289 − 0.891i)12-s + (0.301 + 0.928i)13-s + (−0.0924 + 0.284i)14-s + (0.260 + 0.801i)16-s + (0.501 − 0.364i)17-s − 0.00442·18-s + (0.375 − 0.273i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.697598 + 0.529506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.697598 + 0.529506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.100 + 0.310i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.38 + 1.00i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 + (1.65 - 5.07i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.08 - 3.34i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.06 + 1.50i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 1.19i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.33 - 7.20i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.83 + 2.78i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.31 - 0.954i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.00414 - 0.0127i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.99 - 9.20i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + (-5.61 - 4.08i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 1.01i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.00685 + 0.0210i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.21 + 3.72i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.33 + 2.42i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (1.89 + 1.37i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.467 - 1.43i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.345 + 0.250i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.88 - 3.55i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.88 - 5.79i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (12.9 + 9.40i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.08077588189727289634778715543, −9.839251131574885269290361224035, −9.409608522336202101916119404550, −7.72858737526834715397355151287, −7.09987994606409506135644088866, −6.48955207392451499921767932185, −5.55765389964480794195096126634, −4.06703990706387426580098827464, −2.97752177659461677340225271355, −1.65773722714577394668865808362,
0.49665956742229486794641439132, 2.67786384444709217339106639135, 3.72256021686787611570665577523, 5.41469378903065940166948289868, 5.75453909957137325666359527208, 6.48878029714795376902277196809, 7.67998839159973641570266590767, 8.690326780784325354780960998911, 10.00139505777010134814187311805, 10.50212422437981128062324696910