| L(s) = 1 | + (−3.98 − 0.630i)3-s + (4.90 − 0.992i)5-s + (−1.90 + 1.90i)7-s + (6.88 + 2.23i)9-s + (1.35 + 4.16i)11-s + (−3.42 + 1.74i)13-s + (−20.1 + 0.861i)15-s + (−12.2 + 1.93i)17-s + (3.72 − 5.12i)19-s + (8.78 − 6.38i)21-s + (5.73 − 11.2i)23-s + (23.0 − 9.72i)25-s + (6.32 + 3.22i)27-s + (31.4 + 43.3i)29-s + (35.0 + 25.4i)31-s + ⋯ |
| L(s) = 1 | + (−1.32 − 0.210i)3-s + (0.980 − 0.198i)5-s + (−0.272 + 0.272i)7-s + (0.765 + 0.248i)9-s + (0.123 + 0.379i)11-s + (−0.263 + 0.134i)13-s + (−1.34 + 0.0574i)15-s + (−0.719 + 0.113i)17-s + (0.195 − 0.269i)19-s + (0.418 − 0.303i)21-s + (0.249 − 0.489i)23-s + (0.921 − 0.389i)25-s + (0.234 + 0.119i)27-s + (1.08 + 1.49i)29-s + (1.12 + 0.820i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.10116 + 0.259306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10116 + 0.259306i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.90 + 0.992i)T \) |
| good | 3 | \( 1 + (3.98 + 0.630i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (1.90 - 1.90i)T - 49iT^{2} \) |
| 11 | \( 1 + (-1.35 - 4.16i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (3.42 - 1.74i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (12.2 - 1.93i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-3.72 + 5.12i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-5.73 + 11.2i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-31.4 - 43.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-35.0 - 25.4i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-21.5 - 42.3i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-19.4 + 59.9i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-7.89 - 7.89i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (0.830 - 5.24i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (21.9 + 3.48i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-86.6 - 28.1i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-4.43 - 13.6i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-11.7 + 1.86i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-103. + 75.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-30.5 + 59.9i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-25.8 - 35.5i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-9.74 - 61.5i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (146. - 47.4i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (2.23 - 14.1i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−11.04150817645735391366429911909, −10.39276485794931344880364261990, −9.440641594320142058385275879421, −8.523131398848503436731502291498, −6.82728234137984449636971589559, −6.45958946182173738539618699502, −5.34750206042595207583783127277, −4.66315481980641619013318349053, −2.64235011366279327212841613075, −1.08527026186864091212429737045,
0.72879911112281769361665339697, 2.56592013196222232841902465764, 4.28183685642653905054703390926, 5.37245136940918224787290839576, 6.15265657093594966027841952285, 6.80677095431685397277656372076, 8.219579639004167223284747720475, 9.616283529452596200193744902858, 10.06582411327841930649360784658, 11.09579521609828984751330283621
