| L(s) = 1 | + (−0.0781 − 0.443i)3-s + (−0.766 + 0.642i)5-s + (−1.06 + 1.85i)7-s + (2.62 − 0.956i)9-s + (2.25 + 3.90i)11-s + (−0.155 + 0.881i)13-s + (0.344 + 0.289i)15-s + (−0.00854 − 0.00311i)17-s + (4.31 + 0.640i)19-s + (0.904 + 0.329i)21-s + (4.94 + 4.15i)23-s + (0.173 − 0.984i)25-s + (−1.30 − 2.25i)27-s + (−0.381 + 0.138i)29-s + (−0.920 + 1.59i)31-s + ⋯ |
| L(s) = 1 | + (−0.0451 − 0.255i)3-s + (−0.342 + 0.287i)5-s + (−0.404 + 0.700i)7-s + (0.876 − 0.318i)9-s + (0.680 + 1.17i)11-s + (−0.0430 + 0.244i)13-s + (0.0889 + 0.0746i)15-s + (−0.00207 − 0.000754i)17-s + (0.989 + 0.146i)19-s + (0.197 + 0.0718i)21-s + (1.03 + 0.865i)23-s + (0.0347 − 0.196i)25-s + (−0.250 − 0.434i)27-s + (−0.0708 + 0.0257i)29-s + (−0.165 + 0.286i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22678 + 0.438313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22678 + 0.438313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (-4.31 - 0.640i)T \) |
| good | 3 | \( 1 + (0.0781 + 0.443i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (1.06 - 1.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.155 - 0.881i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.00854 + 0.00311i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.94 - 4.15i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.381 - 0.138i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.920 - 1.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 + (1.24 + 7.04i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.04 + 0.880i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.13 + 0.413i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.90 - 1.59i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-8.35 - 3.04i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (4.44 + 3.72i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (14.6 - 5.34i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.71 - 4.79i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.93 + 10.9i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.27 + 7.25i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.0758 - 0.131i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.98 + 11.2i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (7.85 + 2.85i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.78669666291027759581886092750, −10.45476251512587561331414031530, −9.553299380245475294013692540564, −8.899093114686662012386118699942, −7.28970619207958993076363165802, −7.05249126655435471215769433283, −5.72564150636529733739266853293, −4.45310863589495463825147721724, −3.27308913539597354435502631572, −1.66553606194610017337958970968,
1.01810927629532665169763914432, 3.22062803019955149577220314178, 4.18209623229230891325348957680, 5.28963227582470402370111960856, 6.61644120724053217610765209598, 7.44332696906657604894941345705, 8.540772769176387001770803732592, 9.462461052465032709674733061805, 10.40171169294690492430876760246, 11.14991870319805255766524532444
