| L(s) = 1 | + (0.915 − 2.09i)2-s + (−2.20 − 2.37i)4-s + (1.59 + 0.240i)5-s + (2.61 + 2.43i)7-s + (−2.67 + 0.934i)8-s + (1.96 − 3.12i)10-s + (2.89 − 2.49i)11-s + (−2.15 + 3.15i)13-s + (7.49 − 3.27i)14-s + (−0.000114 + 0.00152i)16-s + (2.40 + 2.40i)17-s + (3.94 + 2.48i)19-s + (−2.94 − 4.31i)20-s + (−2.57 − 8.35i)22-s + (−1.21 − 0.476i)23-s + ⋯ |
| L(s) = 1 | + (0.647 − 1.48i)2-s + (−1.10 − 1.18i)4-s + (0.713 + 0.107i)5-s + (0.990 + 0.918i)7-s + (−0.944 + 0.330i)8-s + (0.620 − 0.988i)10-s + (0.873 − 0.751i)11-s + (−0.596 + 0.874i)13-s + (2.00 − 0.874i)14-s + (−2.86e−5 + 0.000382i)16-s + (0.582 + 0.582i)17-s + (0.905 + 0.569i)19-s + (−0.657 − 0.964i)20-s + (−0.549 − 1.78i)22-s + (−0.253 − 0.0993i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74072 - 2.02219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74072 - 2.02219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (5.35 - 0.564i)T \) |
| good | 2 | \( 1 + (-0.915 + 2.09i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (-1.59 - 0.240i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-2.61 - 2.43i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.89 + 2.49i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.15 - 3.15i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 2.40i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.94 - 2.48i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (1.21 + 0.476i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-5.64 + 4.16i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (3.43 + 9.83i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (3.98 + 1.06i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.24 - 5.74i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (1.51 + 1.75i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.15 - 0.918i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (10.0 - 5.78i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.11 + 0.0417i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (-1.39 + 0.104i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (11.9 - 5.73i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.714 + 6.34i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (1.64 - 8.71i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.03 + 6.58i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.47 + 0.278i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (2.31 + 4.37i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.16708541166503600480120877192, −9.502283521693533429593115285058, −8.747801768670068174813698842555, −7.58633922313132145338408383544, −6.05522681733131673301823721909, −5.45022340501478135536466228710, −4.39998375291497684549849614370, −3.40922115441072293684513978839, −2.16678442790225787372095073359, −1.51611896538938907644895048102,
1.52701621433972264840814077031, 3.44652831473900754262373985721, 4.76033594885442338994009802995, 5.06238624845575030575319736972, 6.14871380855434365463260630557, 7.16151720672992698123466170445, 7.56218059076761257475865290819, 8.477738640405248162635616672568, 9.627475583085104207743347998187, 10.26835736817783184317374841933
