| L(s) = 1 | + (−1.30 + 0.541i)2-s + (−0.475 + 2.38i)3-s + (1.41 − 1.41i)4-s + (1.85 − 1.24i)5-s + (−0.671 − 3.37i)6-s + (2.54 + 1.70i)7-s + (−1.08 + 2.61i)8-s + (2.83 + 1.17i)9-s + (−1.75 + 2.62i)10-s + (6.67 − 1.32i)11-s + (2.70 + 4.04i)12-s + (2.36 + 2.36i)13-s + (−4.25 − 0.846i)14-s + (2.08 + 5.03i)15-s − 4i·16-s + (−8.63 + 14.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.653 + 0.270i)2-s + (−0.158 + 0.796i)3-s + (0.353 − 0.353i)4-s + (0.371 − 0.248i)5-s + (−0.111 − 0.562i)6-s + (0.364 + 0.243i)7-s + (−0.135 + 0.326i)8-s + (0.315 + 0.130i)9-s + (−0.175 + 0.262i)10-s + (0.606 − 0.120i)11-s + (0.225 + 0.337i)12-s + (0.181 + 0.181i)13-s + (−0.303 − 0.0604i)14-s + (0.138 + 0.335i)15-s − 0.250i·16-s + (−0.508 + 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0834 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0834 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.858004 + 0.789170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858004 + 0.789170i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.541i)T \) |
| 5 | \( 1 + (-1.85 + 1.24i)T \) |
| 17 | \( 1 + (8.63 - 14.6i)T \) |
| good | 3 | \( 1 + (0.475 - 2.38i)T + (-8.31 - 3.44i)T^{2} \) |
| 7 | \( 1 + (-2.54 - 1.70i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-6.67 + 1.32i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-2.36 - 2.36i)T + 169iT^{2} \) |
| 19 | \( 1 + (8.88 - 3.67i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-8.12 - 40.8i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (2.13 + 3.19i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-22.7 - 4.53i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-9.23 + 46.4i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (19.8 + 13.2i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-78.7 - 32.6i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (22.2 + 22.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (19.3 - 8.00i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-1.90 + 4.59i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-14.4 + 21.6i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 96.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-1.45 + 7.31i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-18.6 + 12.4i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-75.9 + 15.1i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-21.9 - 52.9i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-14.6 + 14.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (93.7 + 140. i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.72340424718253156505100000228, −11.41841284626079601758121031437, −10.66465626542246631126428085940, −9.612351977791411729421146029298, −8.946197088650925410188708660379, −7.75915158428345160309457572696, −6.39016614997220240874659091801, −5.25484259252688064854910453856, −3.95282246659550943920379488244, −1.71047234778185702384689468409,
1.00957278855511259637245798647, 2.50206458363822016730488625180, 4.44546752192672918736011020354, 6.36034076333689658094049817502, 7.01020155331230632611641127745, 8.176533592178234200038974599990, 9.272220115319516716599689267651, 10.33675161206434468287779903340, 11.28563114932148268988431482000, 12.24019000493227482639529331262
