| L(s) = 1 | + (2.70 − 1.37i)2-s + (−4.42 + 0.701i)3-s + (3.04 − 4.19i)4-s + (1.95 + 4.60i)5-s + (−10.9 + 7.98i)6-s + (−4.77 − 4.77i)7-s + (0.561 − 3.54i)8-s + (10.5 − 3.42i)9-s + (11.6 + 9.74i)10-s + (3.84 − 11.8i)11-s + (−10.5 + 20.7i)12-s + (−1.05 − 0.536i)13-s + (−19.4 − 6.32i)14-s + (−11.8 − 19.0i)15-s + (3.04 + 9.37i)16-s + (−4.59 − 0.727i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.687i)2-s + (−1.47 + 0.233i)3-s + (0.761 − 1.04i)4-s + (0.390 + 0.920i)5-s + (−1.83 + 1.33i)6-s + (−0.682 − 0.682i)7-s + (0.0701 − 0.443i)8-s + (1.17 − 0.380i)9-s + (1.16 + 0.974i)10-s + (0.349 − 1.07i)11-s + (−0.879 + 1.72i)12-s + (−0.0809 − 0.0412i)13-s + (−1.39 − 0.451i)14-s + (−0.790 − 1.26i)15-s + (0.190 + 0.586i)16-s + (−0.270 − 0.0428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14922 - 0.304370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14922 - 0.304370i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.95 - 4.60i)T \) |
| good | 2 | \( 1 + (-2.70 + 1.37i)T + (2.35 - 3.23i)T^{2} \) |
| 3 | \( 1 + (4.42 - 0.701i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (4.77 + 4.77i)T + 49iT^{2} \) |
| 11 | \( 1 + (-3.84 + 11.8i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (1.05 + 0.536i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (4.59 + 0.727i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-12.8 - 17.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (4.03 + 7.92i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-5.42 + 7.47i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (25.3 - 18.3i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-6.47 + 12.7i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (16.8 + 51.9i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (36.1 - 36.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (0.703 + 4.43i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-69.2 + 10.9i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (67.9 - 22.0i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-15.0 + 46.4i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-79.5 - 12.5i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (34.9 + 25.3i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-27.3 - 53.6i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (27.4 - 37.8i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-17.1 + 108. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-63.0 - 20.4i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-8.99 - 56.7i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−17.14515420787957870369660951848, −16.06480842248407120889378601851, −14.40137502760945081317708592455, −13.40728468789511859032026970936, −12.04189602733842473300451450136, −11.04819642520134032161414822415, −10.23951242762117829005256446128, −6.56861192447885104135930906292, −5.55400752615942619827045375248, −3.62115111574383181618291944398,
4.75238264214213962287153070750, 5.75288200614606951911148458013, 6.89036643746500892253926257780, 9.582404787379575785360509344800, 11.79163291488912233495268733312, 12.52892855135288227940506605678, 13.40101627008352333532351247125, 15.21130472555166873605888480452, 16.19899460296504418336138653913, 17.08526460403164668439191811256
