L(s) = 1 | + (0.366 − 1.36i)2-s + (−1.52 − 2.63i)3-s + (−1.73 − i)4-s + (4.79 + 4.79i)5-s + (−4.15 + 1.11i)6-s + (1.13 + 4.25i)7-s + (−2 + 1.99i)8-s + (−0.132 + 0.228i)9-s + (8.29 − 4.79i)10-s + (−13.8 − 3.71i)11-s + 6.08i·12-s + (1.84 + 12.8i)13-s + 6.22·14-s + (5.33 − 19.9i)15-s + (1.99 + 3.46i)16-s + (−20.9 − 12.1i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.507 − 0.878i)3-s + (−0.433 − 0.250i)4-s + (0.958 + 0.958i)5-s + (−0.692 + 0.185i)6-s + (0.162 + 0.607i)7-s + (−0.250 + 0.249i)8-s + (−0.0146 + 0.0254i)9-s + (0.829 − 0.479i)10-s + (−1.26 − 0.337i)11-s + 0.507i·12-s + (0.142 + 0.989i)13-s + 0.444·14-s + (0.355 − 1.32i)15-s + (0.124 + 0.216i)16-s + (−1.23 − 0.713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.797163 - 0.504088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.797163 - 0.504088i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 13 | \( 1 + (-1.84 - 12.8i)T \) |
good | 3 | \( 1 + (1.52 + 2.63i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-4.79 - 4.79i)T + 25iT^{2} \) |
| 7 | \( 1 + (-1.13 - 4.25i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (13.8 + 3.71i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (20.9 + 12.1i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-25.4 + 6.82i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (5.44 - 3.14i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-11.1 - 19.2i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (8.59 + 8.59i)T + 961iT^{2} \) |
| 37 | \( 1 + (6.13 + 1.64i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-18.8 + 70.4i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-26.9 - 15.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-7.65 + 7.65i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 33.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-9.77 - 36.4i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-11.5 + 19.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (27.8 - 103. i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-2.20 + 0.591i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-38.1 + 38.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 19.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-34.7 - 34.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-3.47 - 0.930i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (24.3 - 6.51i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−17.83395785318610574556746966477, −15.74021765406459828305497481496, −14.06424256204560219922896248060, −13.27179097485458759740203114433, −11.87325995391701706659798274239, −10.81599139316149899503033178225, −9.296813006306228678138216179947, −7.00799684015701738839241920865, −5.59585237339799816418015853747, −2.39308500621309395742667048041,
4.67875794261568990535766948499, 5.68351322626094566961703263013, 7.938031072020424465250746126613, 9.649245623079041604553375536497, 10.67389132930586184320321905965, 12.84736499239125833691645533044, 13.65601544971397662628606055556, 15.42908404413254002415913698509, 16.21283475162627782321747990260, 17.27905456444789530368112102224