| L(s) = 1 | + (−0.303 + 0.478i)2-s + (1.56 − 1.47i)3-s + (0.715 + 1.51i)4-s + (−0.783 + 2.09i)5-s + (0.228 + 1.19i)6-s + (0.694 − 0.504i)7-s + (−2.06 − 0.261i)8-s + (0.102 − 1.62i)9-s + (−0.763 − 1.00i)10-s + (0.598 − 0.943i)11-s + (3.35 + 1.33i)12-s + (0.437 − 6.95i)13-s + (0.0304 + 0.484i)14-s + (1.85 + 4.43i)15-s + (−1.38 + 1.67i)16-s + (−1.37 + 2.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.214 + 0.338i)2-s + (0.905 − 0.850i)3-s + (0.357 + 0.759i)4-s + (−0.350 + 0.936i)5-s + (0.0931 + 0.488i)6-s + (0.262 − 0.190i)7-s + (−0.730 − 0.0923i)8-s + (0.0341 − 0.542i)9-s + (−0.241 − 0.319i)10-s + (0.180 − 0.284i)11-s + (0.969 + 0.383i)12-s + (0.121 − 1.92i)13-s + (0.00815 + 0.129i)14-s + (0.479 + 1.14i)15-s + (−0.347 + 0.419i)16-s + (−0.332 + 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21776 + 0.276506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21776 + 0.276506i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.783 - 2.09i)T \) |
| good | 2 | \( 1 + (0.303 - 0.478i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (-1.56 + 1.47i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-0.694 + 0.504i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.598 + 0.943i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (-0.437 + 6.95i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (1.37 - 2.91i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (1.40 + 1.31i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.0964 + 0.0247i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (4.73 + 2.60i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (1.40 - 2.98i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-6.42 + 7.76i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (6.77 - 1.73i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (1.85 - 5.69i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-1.51 + 0.190i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (0.347 - 1.81i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (-13.6 - 5.39i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-8.33 - 2.13i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (9.67 - 5.31i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (6.46 - 0.816i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-6.22 + 2.46i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-9.40 + 8.83i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (7.63 + 7.17i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-8.34 + 3.30i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-6.42 - 3.53i)T + (51.9 + 81.8i)T^{2} \) |
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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
−13.30122437102915590982685924907, −12.75767646016623271865613736249, −11.42869398599792959416822034153, −10.48088699441532498955554353269, −8.698809319936887236800475295627, −7.88324049917997907590240447619, −7.34404618548684533788467011842, −6.12555002183583682542212223701, −3.58411970554032984014686050263, −2.55152506417885192522645625997,
1.98242734921298904516066835942, 3.94537355993545630718491097829, 5.05027944638877897904162587918, 6.75989706509425436894565639301, 8.491722399543920953688776509571, 9.251058700832893073386105708825, 9.837954840456363808281838522688, 11.33685862669563509212780593830, 11.95516193494901973338313452867, 13.53178480829496132873823939445
