| L(s) = 1 | + (1.13 + 2.74i)2-s + (−1.44 + 0.962i)3-s + (−3.39 + 3.39i)4-s + (2.36 + 0.469i)5-s + (−4.27 − 2.85i)6-s + (−3.45 + 0.687i)7-s + (−2.19 − 0.908i)8-s + (1.14 − 2.77i)9-s + (1.39 + 7.00i)10-s + (9.01 − 13.4i)11-s + (1.62 − 8.15i)12-s + (1.90 + 1.90i)13-s + (−5.80 − 8.69i)14-s + (−3.85 + 1.59i)15-s + 12.1i·16-s + (15.6 + 6.64i)17-s + ⋯ |
| L(s) = 1 | + (0.567 + 1.37i)2-s + (−0.480 + 0.320i)3-s + (−0.848 + 0.848i)4-s + (0.472 + 0.0939i)5-s + (−0.712 − 0.475i)6-s + (−0.493 + 0.0981i)7-s + (−0.274 − 0.113i)8-s + (0.127 − 0.307i)9-s + (0.139 + 0.700i)10-s + (0.819 − 1.22i)11-s + (0.135 − 0.679i)12-s + (0.146 + 0.146i)13-s + (−0.414 − 0.620i)14-s + (−0.256 + 0.106i)15-s + 0.759i·16-s + (0.920 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.730124 + 1.13592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.730124 + 1.13592i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 - 0.962i)T \) |
| 17 | \( 1 + (-15.6 - 6.64i)T \) |
| good | 2 | \( 1 + (-1.13 - 2.74i)T + (-2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (-2.36 - 0.469i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (3.45 - 0.687i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-9.01 + 13.4i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 1.90i)T + 169iT^{2} \) |
| 19 | \( 1 + (1.93 + 4.67i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (32.9 + 21.9i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-0.0486 + 0.244i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (24.5 + 36.7i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (8.38 - 5.60i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-52.7 + 10.4i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (20.7 - 50.2i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-52.9 - 52.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-11.0 - 26.7i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (10.2 + 4.26i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (5.87 + 29.5i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 106. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-2.20 + 1.47i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-115. - 22.9i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (66.4 - 99.5i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (38.6 - 16.0i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (7.13 - 7.13i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (6.59 - 33.1i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−15.79775868623870750377132115830, −14.49191759104210370676415824605, −13.82319931935683695148516405513, −12.49689937279175749204311111784, −10.98688215661091530477769060960, −9.505918358950653993814639153346, −8.029056251638613110316715366701, −6.30076630284928254782840668094, −5.85192085358886579991377341441, −4.02094896274649223532009141866,
1.76559544269202277957125471014, 3.83365811219061514106664237340, 5.52401799975986202361016139356, 7.23905819570941787679627814283, 9.563857738295138845622806432262, 10.30850430436509021126201300118, 11.78794609494584412546412724106, 12.36511981660482971728996650954, 13.41844688173228913157206452616, 14.39625678393530568746913537916
