| L(s) = 1 | + (0.483 + 1.32i)2-s + (−1.53 + 1.28i)4-s + (−7.52 + 1.32i)5-s + (4.40 + 3.69i)7-s + (−2.44 − 1.41i)8-s + (−5.40 − 9.35i)10-s + (−8.02 − 1.41i)11-s + (−22.0 − 8.02i)13-s + (−2.78 + 7.64i)14-s + (0.694 − 3.93i)16-s + (−6.39 + 3.69i)17-s + (−7.80 + 13.5i)19-s + (9.82 − 11.7i)20-s + (−2.00 − 11.3i)22-s + (19.9 + 23.8i)23-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.383 + 0.321i)4-s + (−1.50 + 0.265i)5-s + (0.629 + 0.528i)7-s + (−0.306 − 0.176i)8-s + (−0.540 − 0.935i)10-s + (−0.729 − 0.128i)11-s + (−1.69 − 0.617i)13-s + (−0.198 + 0.546i)14-s + (0.0434 − 0.246i)16-s + (−0.376 + 0.217i)17-s + (−0.410 + 0.711i)19-s + (0.491 − 0.585i)20-s + (−0.0909 − 0.515i)22-s + (0.868 + 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0536046 - 0.524963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0536046 - 0.524963i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 - 1.32i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (7.52 - 1.32i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.40 - 3.69i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (8.02 + 1.41i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (22.0 + 8.02i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (6.39 - 3.69i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (7.80 - 13.5i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-19.9 - 23.8i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-9.68 - 26.6i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-12.2 + 10.2i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-5.99 - 10.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (2.82 - 7.76i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (7.82 - 44.3i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-43.4 + 51.7i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 16.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (57.3 - 10.1i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-54.6 - 45.8i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (47.1 + 17.1i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (35.4 - 20.4i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (49.7 - 86.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (72.4 - 26.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (36.0 + 99.1i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-0.302 - 0.174i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 62.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.05292266844272751874745848358, −12.17984732034835424617456306886, −11.40842957806401609831476428617, −10.20004423346863681728597562521, −8.645516575517505728939899040366, −7.81147500240728168574389120512, −7.15920892762303890996300847707, −5.44441967934657788830281037051, −4.47319276917107690516235856889, −2.96751981334847991691857823725,
0.29471203040108868466478866246, 2.61855057859497439494329598818, 4.37502226723569762904431854185, 4.80017248181638221933158008503, 7.04916678191920289544937659512, 7.898229947735884278673805227933, 9.026219749348223006281969217976, 10.39743500011622383488066813926, 11.24350121741064573077433517557, 12.05296861184962034385076349919
