| L(s) = 1 | + (−1.78 − 0.579i)2-s + (0.541 − 2.54i)3-s + (1.22 + 0.888i)4-s + (−2.14 − 0.616i)5-s + (−2.43 + 4.22i)6-s + (−0.941 − 2.11i)7-s + (0.538 + 0.741i)8-s + (−3.45 − 1.53i)9-s + (3.47 + 2.34i)10-s + (−0.0632 + 0.601i)11-s + (2.92 − 2.63i)12-s + (3.04 + 2.74i)13-s + (0.453 + 4.31i)14-s + (−2.73 + 5.14i)15-s + (−1.46 − 4.50i)16-s + (−5.41 + 0.568i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 − 0.409i)2-s + (0.312 − 1.47i)3-s + (0.611 + 0.444i)4-s + (−0.961 − 0.275i)5-s + (−0.996 + 1.72i)6-s + (−0.355 − 0.798i)7-s + (0.190 + 0.261i)8-s + (−1.15 − 0.512i)9-s + (1.09 + 0.741i)10-s + (−0.0190 + 0.181i)11-s + (0.844 − 0.760i)12-s + (0.845 + 0.761i)13-s + (0.121 + 1.15i)14-s + (−0.706 + 1.32i)15-s + (−0.366 − 1.12i)16-s + (−1.31 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0572757 + 0.396143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0572757 + 0.396143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.14 + 0.616i)T \) |
| 31 | \( 1 + (-2.81 + 4.80i)T \) |
| good | 2 | \( 1 + (1.78 + 0.579i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.541 + 2.54i)T + (-2.74 - 1.22i)T^{2} \) |
| 7 | \( 1 + (0.941 + 2.11i)T + (-4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.0632 - 0.601i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-3.04 - 2.74i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (5.41 - 0.568i)T + (16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (1.34 + 1.49i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.97 + 4.09i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.285 + 0.878i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (3.46 + 1.99i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.17 - 1.52i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-5.50 + 4.95i)T + (4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-12.1 + 3.96i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.74 + 10.6i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (9.94 + 2.11i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + (12.4 - 7.20i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.4 - 4.66i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.0650 - 0.00684i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.597 - 5.68i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.48 - 6.99i)T + (-75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-7.55 - 5.48i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.33 - 1.83i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.23435291966019822575971031789, −11.37083520103152703101166509405, −10.45132650686542445345706239332, −8.918916611751350836918410943758, −8.362136305250647367028442212485, −7.34215312739694754855320875268, −6.63114517983945530159404623441, −4.18417015422070300966815973308, −2.12602966027875726172305151206, −0.56276690859430038440103624889,
3.26097467119204126326674837781, 4.42165901497725568015405490236, 6.15309834553878953061081763957, 7.65441970171278118349277438565, 8.744637552781505724687587613513, 9.083269047787277533982028117254, 10.41927184412306393040741699577, 10.85191873401021414066236884222, 12.21498067504714426350112575493, 13.69680567037428175214752530701
