| L(s) = 1 | + (2.87 − 0.841i)3-s + (−10.3 − 5.98i)7-s + (7.58 − 4.84i)9-s + (7.43 + 4.29i)11-s + (−14.4 + 8.35i)13-s − 22.1·17-s − 26.9·19-s + (−34.9 − 8.51i)21-s + (19.3 + 33.6i)23-s + (17.7 − 20.3i)27-s + (−8.44 − 4.87i)29-s + (6.01 + 10.4i)31-s + (25.0 + 6.10i)33-s + 25.0i·37-s + (−34.6 + 36.2i)39-s + ⋯ |
| L(s) = 1 | + (0.959 − 0.280i)3-s + (−1.48 − 0.855i)7-s + (0.842 − 0.538i)9-s + (0.675 + 0.390i)11-s + (−1.11 + 0.642i)13-s − 1.30·17-s − 1.41·19-s + (−1.66 − 0.405i)21-s + (0.843 + 1.46i)23-s + (0.657 − 0.753i)27-s + (−0.291 − 0.168i)29-s + (0.194 + 0.336i)31-s + (0.758 + 0.184i)33-s + 0.677i·37-s + (−0.887 + 0.928i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1982378678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1982378678\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.87 + 0.841i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (10.3 + 5.98i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.43 - 4.29i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (14.4 - 8.35i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 22.1T + 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 + (-19.3 - 33.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (8.44 + 4.87i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-6.01 - 10.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 25.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.8 + 20.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (67.8 + 39.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (37.1 - 64.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 13.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (32.9 - 18.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4.56 - 7.90i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-20.8 + 12.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 43.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.68iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (19.3 - 33.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-42.1 + 73.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 61.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (54.2 + 31.3i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.922403894042606196449384701624, −9.424317432699379431719030666845, −8.793664514573851507591591451449, −7.53575805163274257362704926739, −6.83594689368517203954001289082, −6.44950539954615265593499343682, −4.59745500369537719977154884889, −3.86674721548239450853565603982, −2.87179349280870879822609804541, −1.71056534860389521891663092581,
0.05129940722198945327255973646, 2.28420112159386954873305436356, 2.89813517420158406027526687866, 3.98979682318779809406392647621, 5.00143587028042036642803054322, 6.41311613435671223019414456384, 6.79518724975409672916171454060, 8.171860147721750218149871825481, 8.864412800096809049289051547026, 9.419583541686570422783309587713
