| L(s) = 1 | + (−2.93 − 1.49i)2-s + (4.01 + 5.52i)4-s + (0.890 − 4.91i)5-s + (4.34 − 4.34i)7-s + (−1.45 − 9.19i)8-s + (−9.96 + 13.0i)10-s + (4.51 + 13.8i)11-s + (6.86 − 3.49i)13-s + (−19.2 + 6.24i)14-s + (−1.02 + 3.14i)16-s + (25.6 − 4.06i)17-s + (−1.24 + 1.70i)19-s + (30.7 − 14.8i)20-s + (7.51 − 47.4i)22-s + (2.15 − 4.22i)23-s + ⋯ |
| L(s) = 1 | + (−1.46 − 0.746i)2-s + (1.00 + 1.38i)4-s + (0.178 − 0.983i)5-s + (0.620 − 0.620i)7-s + (−0.181 − 1.14i)8-s + (−0.996 + 1.30i)10-s + (0.410 + 1.26i)11-s + (0.528 − 0.269i)13-s + (−1.37 + 0.446i)14-s + (−0.0638 + 0.196i)16-s + (1.50 − 0.238i)17-s + (−0.0653 + 0.0899i)19-s + (1.53 − 0.741i)20-s + (0.341 − 2.15i)22-s + (0.0936 − 0.183i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.502940 - 0.678798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.502940 - 0.678798i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.890 + 4.91i)T \) |
| good | 2 | \( 1 + (2.93 + 1.49i)T + (2.35 + 3.23i)T^{2} \) |
| 7 | \( 1 + (-4.34 + 4.34i)T - 49iT^{2} \) |
| 11 | \( 1 + (-4.51 - 13.8i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-6.86 + 3.49i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-25.6 + 4.06i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (1.24 - 1.70i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-2.15 + 4.22i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (32.6 + 44.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-27.1 - 19.7i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (18.7 + 36.7i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 3.71i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (32.8 + 32.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-12.8 + 81.3i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-81.1 - 12.8i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-10.7 - 3.47i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (9.84 + 30.3i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (0.351 - 0.0556i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (88.5 - 64.3i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (20.1 - 39.4i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-33.1 - 45.6i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-2.71 - 17.1i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-85.2 + 27.7i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-1.37 + 8.70i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−11.72983723032596651665490155912, −10.40719485062878194831400247092, −9.873593317615329417346187705196, −8.913186593894417629068437617379, −8.019163835340047000646705578078, −7.24788229735334478151314271115, −5.37096210593424966471623822617, −3.95452279761355536186516770377, −1.95470749811779147512074858186, −0.869701290023706786294430668657,
1.41992457062934944798777449207, 3.31081098746085985408727464157, 5.64890541257571459236999722489, 6.37144347254091468186054851536, 7.51662932758457342365029936333, 8.359957636854416136598679945272, 9.175042798168438670629132160815, 10.22240795205427790328902636681, 11.04106611129768713878576454426, 11.80941875640266889639852879512
