L(s) = 1 | + (0.0361 + 0.320i)2-s + (3.79 − 0.866i)4-s + (−1.11 − 0.889i)5-s + (−2.06 + 9.04i)7-s + (0.841 + 2.40i)8-s + (0.245 − 0.390i)10-s + (−3.02 + 8.63i)11-s + (7.25 + 15.0i)13-s + (−2.97 − 0.335i)14-s + (13.2 − 6.40i)16-s + (13.4 − 13.4i)17-s + (−6.13 − 3.85i)19-s + (−5.00 − 2.41i)20-s + (−2.87 − 0.657i)22-s + (26.9 + 33.8i)23-s + ⋯ |
L(s) = 1 | + (0.0180 + 0.160i)2-s + (0.949 − 0.216i)4-s + (−0.223 − 0.177i)5-s + (−0.294 + 1.29i)7-s + (0.105 + 0.300i)8-s + (0.0245 − 0.0390i)10-s + (−0.274 + 0.785i)11-s + (0.558 + 1.15i)13-s + (−0.212 − 0.0239i)14-s + (0.831 − 0.400i)16-s + (0.793 − 0.793i)17-s + (−0.322 − 0.202i)19-s + (−0.250 − 0.120i)20-s + (−0.130 − 0.0298i)22-s + (1.17 + 1.47i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61780 + 0.905603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61780 + 0.905603i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 + (8.81 + 27.6i)T \) |
good | 2 | \( 1 + (-0.0361 - 0.320i)T + (-3.89 + 0.890i)T^{2} \) |
| 5 | \( 1 + (1.11 + 0.889i)T + (5.56 + 24.3i)T^{2} \) |
| 7 | \( 1 + (2.06 - 9.04i)T + (-44.1 - 21.2i)T^{2} \) |
| 11 | \( 1 + (3.02 - 8.63i)T + (-94.6 - 75.4i)T^{2} \) |
| 13 | \( 1 + (-7.25 - 15.0i)T + (-105. + 132. i)T^{2} \) |
| 17 | \( 1 + (-13.4 + 13.4i)T - 289iT^{2} \) |
| 19 | \( 1 + (6.13 + 3.85i)T + (156. + 325. i)T^{2} \) |
| 23 | \( 1 + (-26.9 - 33.8i)T + (-117. + 515. i)T^{2} \) |
| 31 | \( 1 + (-2.27 - 20.1i)T + (-936. + 213. i)T^{2} \) |
| 37 | \( 1 + (-5.54 - 15.8i)T + (-1.07e3 + 853. i)T^{2} \) |
| 41 | \( 1 + (-11.0 - 11.0i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (1.43 + 0.161i)T + (1.80e3 + 411. i)T^{2} \) |
| 47 | \( 1 + (52.6 + 18.4i)T + (1.72e3 + 1.37e3i)T^{2} \) |
| 53 | \( 1 + (26.2 - 32.9i)T + (-625. - 2.73e3i)T^{2} \) |
| 59 | \( 1 - 40.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + (31.5 + 50.2i)T + (-1.61e3 + 3.35e3i)T^{2} \) |
| 67 | \( 1 + (-27.0 + 56.0i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (16.5 + 34.2i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-10.1 + 90.2i)T + (-5.19e3 - 1.18e3i)T^{2} \) |
| 79 | \( 1 + (68.6 - 24.0i)T + (4.87e3 - 3.89e3i)T^{2} \) |
| 83 | \( 1 + (14.4 + 63.1i)T + (-6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (-3.18 - 28.2i)T + (-7.72e3 + 1.76e3i)T^{2} \) |
| 97 | \( 1 + (-59.0 + 93.9i)T + (-4.08e3 - 8.47e3i)T^{2} \) |
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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
−11.81608383018562437621814432093, −11.26811391035148731417840613378, −9.887553699915504409855760945078, −9.114711169877388852042976983788, −7.900876412205949314295966447669, −6.87862269097468416889041918596, −5.94033633616690053351644180039, −4.86988007424861666453028116602, −3.06123084585353026329727772349, −1.82551315577457884898152239831,
1.01092853525088939243614156224, 3.02935791791910102263146592186, 3.82364060739053956109836541965, 5.65239642725905182970214889383, 6.71880660365670720332647926770, 7.61546244969610407038985269110, 8.445893475977800688153954353412, 10.15247156870725145712497585653, 10.73638329514337696802209425938, 11.29034501476742337968559509814