| L(s) = 1 | + (0.707 + 1.22i)2-s + (0.0646 + 2.99i)3-s + (−0.999 + 1.73i)4-s − 1.88·5-s + (−3.62 + 2.19i)6-s + (4.14 + 2.39i)7-s − 2.82·8-s + (−8.99 + 0.387i)9-s + (−1.33 − 2.31i)10-s + (2.50 + 4.33i)11-s + (−5.25 − 2.88i)12-s + (7.59 − 10.5i)13-s + 6.76i·14-s + (−0.121 − 5.66i)15-s + (−2.00 − 3.46i)16-s + (15.3 + 8.84i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.0215 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.377·5-s + (−0.604 + 0.366i)6-s + (0.591 + 0.341i)7-s − 0.353·8-s + (−0.999 + 0.0430i)9-s + (−0.133 − 0.231i)10-s + (0.227 + 0.393i)11-s + (−0.438 − 0.240i)12-s + (0.583 − 0.811i)13-s + 0.483i·14-s + (−0.00813 − 0.377i)15-s + (−0.125 − 0.216i)16-s + (0.901 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.766802 + 1.19836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766802 + 1.19836i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (-0.0646 - 2.99i)T \) |
| 13 | \( 1 + (-7.59 + 10.5i)T \) |
| good | 5 | \( 1 + 1.88T + 25T^{2} \) |
| 7 | \( 1 + (-4.14 - 2.39i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.50 - 4.33i)T + (-60.5 + 104. i)T^{2} \) |
| 17 | \( 1 + (-15.3 - 8.84i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8.89 - 5.13i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-30.3 + 17.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-5.98 + 3.45i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 30.9iT - 961T^{2} \) |
| 37 | \( 1 + (35.4 - 20.4i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-6.44 - 11.1i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-18.5 + 32.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + 79.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 30.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (34.0 - 59.0i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.6 - 60.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-12.6 + 7.33i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-68.5 + 118. i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + 38.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 100.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + (41.9 + 72.6i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (114. + 65.8i)T + (4.70e3 + 8.14e3i)T^{2} \) |
| 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
−14.96776203393022832737070720434, −13.77960997372775873302264050736, −12.35675730367556251362103333736, −11.29106964112236938305240316601, −10.06947936058067646728239149982, −8.721127912750953313909756047745, −7.77587610562422276602615624526, −5.92721631089684393035727760563, −4.79641655043033006956028619678, −3.41942669586522773798808778224,
1.33727345255295185695065507014, 3.37791158254257610235048561768, 5.22196701760479102155807768083, 6.79870469088660232404791360883, 8.024076343460753608378901404764, 9.306238405089115058289172373506, 11.10967829553113817131798762305, 11.61670423290128110472620315663, 12.72668704882270344649837104670, 13.87910171542583754053066584824
