| L(s) = 1 | + (−0.278 − 1.98i)2-s + (1.28 − 2.71i)3-s + (−3.84 + 1.10i)4-s + (−5.80 + 2.11i)5-s + (−5.72 − 1.79i)6-s + (−7.71 + 1.36i)7-s + (3.25 + 7.30i)8-s + (−5.69 − 6.96i)9-s + (5.80 + 10.9i)10-s + (5.58 − 15.3i)11-s + (−1.95 + 11.8i)12-s + (0.501 + 0.420i)13-s + (4.84 + 14.9i)14-s + (−1.73 + 18.4i)15-s + (13.5 − 8.47i)16-s + (16.2 − 28.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.139 − 0.990i)2-s + (0.428 − 0.903i)3-s + (−0.961 + 0.275i)4-s + (−1.16 + 0.422i)5-s + (−0.954 − 0.298i)6-s + (−1.10 + 0.194i)7-s + (0.406 + 0.913i)8-s + (−0.632 − 0.774i)9-s + (0.580 + 1.09i)10-s + (0.507 − 1.39i)11-s + (−0.163 + 0.986i)12-s + (0.0385 + 0.0323i)13-s + (0.345 + 1.06i)14-s + (−0.115 + 1.23i)15-s + (0.848 − 0.529i)16-s + (0.956 − 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.141640 + 0.594472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.141640 + 0.594472i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.278 + 1.98i)T \) |
| 3 | \( 1 + (-1.28 + 2.71i)T \) |
| good | 5 | \( 1 + (5.80 - 2.11i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (7.71 - 1.36i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-5.58 + 15.3i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-0.501 - 0.420i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-16.2 + 28.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (22.1 - 12.7i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.4 + 2.02i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-0.277 + 0.232i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-17.8 - 3.14i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-16.6 + 28.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.6 - 11.4i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-1.58 + 4.36i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-58.2 + 10.2i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-7.73 - 21.2i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (11.4 + 65.0i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-20.4 + 24.4i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (56.4 + 32.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-27.4 - 47.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-18.8 - 22.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-11.6 - 13.9i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (37.7 + 65.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-144. - 52.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.59299055148097033481048669583, −11.91719928145367972756375220810, −11.07382555954666386717796652100, −9.584415196616674818613870904425, −8.533013055743844874418887229586, −7.55498362457325156366664121879, −6.11577438318277806478011883916, −3.72318960393885221462600146364, −2.89972476152662586627332331761, −0.45169670218375524305698742950,
3.80836938971420821567959079212, 4.50226893428450603461684425742, 6.25595040255687514832547929064, 7.62888085178921465866002989689, 8.556172159546027885986505522813, 9.626460923972051172409046368162, 10.44778045089918720362496525280, 12.25233424059039251783001819154, 13.09578979595689687640853116753, 14.55139401074926776611912402913
