| L(s) = 1 | + (−1.39 + 2.65i)3-s + (2.45 − 1.41i)7-s + (−5.10 − 7.40i)9-s + (−0.949 + 0.547i)11-s + (3.63 + 2.09i)13-s + 7.85·17-s − 13.3·19-s + (0.340 + 8.48i)21-s + (12.3 − 21.4i)23-s + (26.8 − 3.23i)27-s + (2.43 − 1.40i)29-s + (12.0 − 20.8i)31-s + (−0.131 − 3.28i)33-s − 49.9i·37-s + (−10.6 + 6.72i)39-s + ⋯ |
| L(s) = 1 | + (−0.464 + 0.885i)3-s + (0.350 − 0.202i)7-s + (−0.567 − 0.823i)9-s + (−0.0862 + 0.0498i)11-s + (0.279 + 0.161i)13-s + 0.462·17-s − 0.703·19-s + (0.0161 + 0.404i)21-s + (0.537 − 0.931i)23-s + (0.992 − 0.119i)27-s + (0.0840 − 0.0485i)29-s + (0.389 − 0.673i)31-s + (−0.00398 − 0.0995i)33-s − 1.34i·37-s + (−0.272 + 0.172i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.541275168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541275168\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.39 - 2.65i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-2.45 + 1.41i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.949 - 0.547i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.63 - 2.09i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 7.85T + 289T^{2} \) |
| 19 | \( 1 + 13.3T + 361T^{2} \) |
| 23 | \( 1 + (-12.3 + 21.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-2.43 + 1.40i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-12.0 + 20.8i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 49.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (18.9 + 10.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-42.4 + 24.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-33.8 - 58.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-86.8 - 50.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.5 + 71.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-49.8 - 28.7i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 14.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-54.4 - 94.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (40.2 + 69.7i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 65.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-134. + 77.4i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.07218849268577350565339874994, −9.143491467245444673451907260041, −8.459993625490619637739414613129, −7.36945393137750042305032484188, −6.31904564263171158704633279019, −5.51918215862058835119817531395, −4.52220257728183904432829205187, −3.82543713619237273410574177754, −2.48963520208147401414051794110, −0.71846865437920582164909892629,
0.940794649380757344460439334654, 2.09035802800821956840218059396, 3.33360315631330397554875418006, 4.80182454916829037806206761057, 5.58571147918284147381715812990, 6.47727926744428033052154044753, 7.29531703208617595035363767283, 8.170130650369508064758189184350, 8.798247516321569117097699306718, 10.05046980177141131084391166964
