| L(s) = 1 | − 2.82i·2-s − 12.6·3-s − 8.00·4-s + 35.8i·6-s + (−48.5 + 6.45i)7-s + 22.6i·8-s + 79.9·9-s + 191.·11-s + 101.·12-s − 48.5·13-s + (18.2 + 137. i)14-s + 64.0·16-s + 181.·17-s − 226. i·18-s − 599. i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.40·3-s − 0.500·4-s + 0.996i·6-s + (−0.991 + 0.131i)7-s + 0.353i·8-s + 0.987·9-s + 1.58·11-s + 0.704·12-s − 0.287·13-s + (0.0931 + 0.700i)14-s + 0.250·16-s + 0.629·17-s − 0.698i·18-s − 1.66i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2881889453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2881889453\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (48.5 - 6.45i)T \) |
| good | 3 | \( 1 + 12.6T + 81T^{2} \) |
| 11 | \( 1 - 191.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 48.5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 181.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 599. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 469. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 338.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 267. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 668. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.32e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.94e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.93e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.46e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.73e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 246. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.07e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.27e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.10e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.01e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.44e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 2.13e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.89e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.49423044087122203394402040474, −9.612631736082978994640906650373, −8.911865933175532228724248427369, −7.11831069063727770820888971239, −6.36652860477550958534683739958, −5.39133134416769433339951832327, −4.28296523788016589261941946221, −3.05417708760987828559336972163, −1.23983567507322747794766459944, −0.13270559716722082827945837586,
1.14878450057713696975758715938, 3.55597592250143701900998229229, 4.63684127202699085294925778690, 5.93822653821167515039575964256, 6.31076200914345868244279588192, 7.19089792457143802656686690137, 8.517848344489768480224431761502, 9.729998372137406702872350374460, 10.25762395829143144882366640907, 11.54947473631392610363391221574
