| L(s) = 1 | + (−1.23 + 0.693i)2-s + (0.209 + 1.05i)3-s + (1.03 − 1.70i)4-s + (2.15 − 3.23i)5-s + (−0.988 − 1.15i)6-s + (−1.47 − 2.21i)7-s + (−0.0939 + 2.82i)8-s + (1.70 − 0.706i)9-s + (−0.420 + 5.48i)10-s + (−1.73 − 0.345i)11-s + (2.01 + 0.735i)12-s + (2.86 + 2.86i)13-s + (3.36 + 1.70i)14-s + (3.85 + 1.59i)15-s + (−1.84 − 3.54i)16-s + (−3.28 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.871 + 0.490i)2-s + (0.120 + 0.607i)3-s + (0.519 − 0.854i)4-s + (0.965 − 1.44i)5-s + (−0.403 − 0.470i)6-s + (−0.559 − 0.837i)7-s + (−0.0332 + 0.999i)8-s + (0.568 − 0.235i)9-s + (−0.132 + 1.73i)10-s + (−0.522 − 0.104i)11-s + (0.582 + 0.212i)12-s + (0.793 + 0.793i)13-s + (0.898 + 0.455i)14-s + (0.995 + 0.412i)15-s + (−0.461 − 0.887i)16-s + (−0.797 + 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.870654 - 0.0137460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.870654 - 0.0137460i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 0.693i)T \) |
| 17 | \( 1 + (3.28 - 2.48i)T \) |
| good | 3 | \( 1 + (-0.209 - 1.05i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-2.15 + 3.23i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (1.47 + 2.21i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (1.73 + 0.345i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.86 - 2.86i)T + 13iT^{2} \) |
| 19 | \( 1 + (-6.12 - 2.53i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (6.80 + 1.35i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.863 - 0.577i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.529 - 2.66i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (3.45 - 0.687i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (3.35 - 2.24i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.51 + 2.28i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 1.38i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.86 - 4.50i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.98 + 4.79i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.41 - 2.94i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 7.78iT - 67T^{2} \) |
| 71 | \( 1 + (-1.59 + 0.317i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.63 - 2.42i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.07 - 5.38i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (1.56 - 3.78i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.341 - 0.341i)T + 89iT^{2} \) |
| 97 | \( 1 + (8.90 - 13.3i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−13.45857440225969686521661537307, −12.23944260544524717503798885478, −10.60868193515344123207942882395, −9.855792705134942467708832532181, −9.193938686919563676958105704283, −8.211070716156718519945672448332, −6.70123366120217005437907753346, −5.55773982328690276442766504308, −4.21951332712297687496065411566, −1.43041788645144210303775112236,
2.15648430284582964510206294789, 3.09374890031304617786545627241, 5.92726639189689640370379809151, 6.91267259189543184554539232539, 7.84194410003754900397697490687, 9.328329972089889931479481509136, 10.10737432369243549023444788803, 10.96049370550917216653139737715, 12.09828826038808369898939635679, 13.26250936092200011736746444179
