| L(s) = 1 | + (0.887 − 2.53i)2-s + (−2.51 − 2.00i)4-s + (−3.04 − 6.31i)5-s + (−0.484 − 0.608i)7-s + (1.77 − 1.11i)8-s + (−18.7 + 2.10i)10-s + (−4.24 − 2.66i)11-s + (−2.96 + 0.675i)13-s + (−1.97 + 0.690i)14-s + (−4.11 − 18.0i)16-s + (−5.44 + 5.44i)17-s + (1.87 + 16.6i)19-s + (−5.02 + 22.0i)20-s + (−10.5 + 8.40i)22-s + (−16.4 − 7.94i)23-s + ⋯ |
| L(s) = 1 | + (0.443 − 1.26i)2-s + (−0.629 − 0.502i)4-s + (−0.608 − 1.26i)5-s + (−0.0692 − 0.0868i)7-s + (0.221 − 0.139i)8-s + (−1.87 + 0.210i)10-s + (−0.386 − 0.242i)11-s + (−0.227 + 0.0519i)13-s + (−0.140 + 0.0493i)14-s + (−0.257 − 1.12i)16-s + (−0.320 + 0.320i)17-s + (0.0988 + 0.877i)19-s + (−0.251 + 1.10i)20-s + (−0.479 + 0.381i)22-s + (−0.717 − 0.345i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.186874 + 1.48056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.186874 + 1.48056i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (10.4 + 27.0i)T \) |
| good | 2 | \( 1 + (-0.887 + 2.53i)T + (-3.12 - 2.49i)T^{2} \) |
| 5 | \( 1 + (3.04 + 6.31i)T + (-15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (0.484 + 0.608i)T + (-10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (4.24 + 2.66i)T + (52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (2.96 - 0.675i)T + (152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (5.44 - 5.44i)T - 289iT^{2} \) |
| 19 | \( 1 + (-1.87 - 16.6i)T + (-351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (16.4 + 7.94i)T + (329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-7.35 + 21.0i)T + (-751. - 599. i)T^{2} \) |
| 37 | \( 1 + (-6.78 + 4.26i)T + (593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (-35.8 - 35.8i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-61.1 + 21.3i)T + (1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 37.5i)T + (-958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (21.6 - 10.4i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 - 90.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-10.2 - 1.15i)T + (3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (-20.7 - 4.72i)T + (4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (88.7 - 20.2i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-2.59 - 7.42i)T + (-4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-13.3 - 21.2i)T + (-2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-81.3 + 101. i)T + (-1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-34.1 + 97.6i)T + (-6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-113. + 12.7i)T + (9.17e3 - 2.09e3i)T^{2} \) |
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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
−11.53919199054332482654926642061, −10.46894469195366854363377503029, −9.588224098636271555490718484477, −8.435513141879496893647728140152, −7.54709413028619407501258422986, −5.76986413225482683446117816651, −4.50957515071852113173125384397, −3.82393112979905085663628819143, −2.21682406662470943153775197556, −0.66181131167531333376279100581,
2.65865679669143151586571803859, 4.11582088644812397608246214106, 5.33059666340175459009897915403, 6.47613103445682204004185092271, 7.24104581407421809967205119256, 7.83937103374293016153619165581, 9.184192169234621471041592424290, 10.59600467046731985248662081671, 11.16853416798699623080871543178, 12.40511572941677410662932513547
