| L(s) = 1 | + (1.35 + 0.398i)2-s + (−0.802 + 1.24i)3-s + (1.68 + 1.08i)4-s + (1.23 − 0.177i)5-s + (−1.58 + 1.37i)6-s + (1.90 − 2.66i)7-s + (1.85 + 2.13i)8-s + (0.332 + 0.726i)9-s + (1.74 + 0.251i)10-s + (−0.850 − 1.08i)11-s + (−2.69 + 1.23i)12-s + (1.92 + 0.467i)13-s + (3.64 − 2.86i)14-s + (−0.767 + 1.68i)15-s + (1.65 + 3.63i)16-s + (−4.85 + 0.934i)17-s + ⋯ |
| L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.463 + 0.720i)3-s + (0.841 + 0.540i)4-s + (0.551 − 0.0793i)5-s + (−0.647 + 0.560i)6-s + (0.718 − 1.00i)7-s + (0.654 + 0.756i)8-s + (0.110 + 0.242i)9-s + (0.551 + 0.0793i)10-s + (−0.256 − 0.326i)11-s + (−0.779 + 0.355i)12-s + (0.534 + 0.129i)13-s + (0.973 − 0.765i)14-s + (−0.198 + 0.434i)15-s + (0.414 + 0.909i)16-s + (−1.17 + 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28612 + 1.27741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28612 + 1.27741i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 0.398i)T \) |
| 67 | \( 1 + (-7.68 + 2.81i)T \) |
| good | 3 | \( 1 + (0.802 - 1.24i)T + (-1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-1.23 + 0.177i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.90 + 2.66i)T + (-2.28 - 6.61i)T^{2} \) |
| 11 | \( 1 + (0.850 + 1.08i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 0.467i)T + (11.5 + 5.95i)T^{2} \) |
| 17 | \( 1 + (4.85 - 0.934i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-3.55 + 2.52i)T + (6.21 - 17.9i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 1.38i)T + (13.3 - 18.7i)T^{2} \) |
| 29 | \( 1 + (5.89 - 3.40i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.231 + 0.953i)T + (-27.5 + 14.2i)T^{2} \) |
| 37 | \( 1 + (-0.845 - 0.488i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.23 - 3.56i)T + (-32.2 - 25.3i)T^{2} \) |
| 43 | \( 1 + (0.108 - 0.0944i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (0.152 - 3.19i)T + (-46.7 - 4.46i)T^{2} \) |
| 53 | \( 1 + (6.27 + 5.44i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.30 + 7.85i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.33 + 6.78i)T + (-14.3 - 59.2i)T^{2} \) |
| 71 | \( 1 + (9.10 + 1.75i)T + (65.9 + 26.3i)T^{2} \) |
| 73 | \( 1 + (8.29 + 6.51i)T + (17.2 + 70.9i)T^{2} \) |
| 79 | \( 1 + (0.0432 + 0.0412i)T + (3.75 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-5.14 + 12.8i)T + (-60.0 - 57.2i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 7.44i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.73 - 2.99i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09977922510914562426651136910, −10.47855304726688165482629142619, −9.318555569382021070769886402981, −8.061147253150489425873575358978, −7.20142027169287567237494960305, −6.15909723074235572092048666163, −5.13369904628158878003983804930, −4.55745092193279967132789488970, −3.52421139447836267038107388280, −1.84985686361957688100379668235,
1.53309223036949128994887309788, 2.46263836617446455281110573787, 3.99866615861954231640908615065, 5.35466133853978586902632788378, 5.81816734975276499733919179443, 6.78185516673094006698847781906, 7.69220286116554807092240044589, 9.024330228613394937960293777346, 9.969205825763519921919586372743, 11.13428344358027762329148376371
