L(s) = 1 | + (0.914 + 1.07i)2-s + (0.980 − 0.195i)3-s + (−0.325 + 1.97i)4-s + (−1.12 − 1.67i)5-s + (1.10 + 0.879i)6-s + (3.63 + 1.50i)7-s + (−2.42 + 1.45i)8-s + (0.923 − 0.382i)9-s + (0.783 − 2.74i)10-s + (0.181 − 0.913i)11-s + (0.0654 + 1.99i)12-s + (−2.94 + 4.40i)13-s + (1.70 + 5.29i)14-s + (−1.42 − 1.42i)15-s + (−3.78 − 1.28i)16-s + (2.67 − 2.67i)17-s + ⋯ |
L(s) = 1 | + (0.646 + 0.762i)2-s + (0.566 − 0.112i)3-s + (−0.162 + 0.986i)4-s + (−0.501 − 0.750i)5-s + (0.452 + 0.358i)6-s + (1.37 + 0.568i)7-s + (−0.857 + 0.514i)8-s + (0.307 − 0.127i)9-s + (0.247 − 0.867i)10-s + (0.0548 − 0.275i)11-s + (0.0188 + 0.577i)12-s + (−0.816 + 1.22i)13-s + (0.454 + 1.41i)14-s + (−0.368 − 0.368i)15-s + (−0.946 − 0.321i)16-s + (0.647 − 0.647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61566 + 0.863600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61566 + 0.863600i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.914 - 1.07i)T \) |
| 3 | \( 1 + (-0.980 + 0.195i)T \) |
good | 5 | \( 1 + (1.12 + 1.67i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-3.63 - 1.50i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.181 + 0.913i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (2.94 - 4.40i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-2.67 + 2.67i)T - 17iT^{2} \) |
| 19 | \( 1 + (6.32 + 4.22i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (3.18 + 7.69i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.769 - 3.86i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 4.08iT - 31T^{2} \) |
| 37 | \( 1 + (-2.30 + 1.54i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.846 - 2.04i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-3.25 - 0.647i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.22 - 11.1i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (4.86 + 7.27i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (7.71 - 1.53i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-10.8 + 2.15i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (8.29 + 3.43i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (1.40 - 0.581i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-7.55 - 7.55i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.83 - 2.56i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (2.90 - 7.01i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 5.94iT - 97T^{2} \) |
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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
−12.53955725867763938289152659832, −12.17031926550466242159992490809, −11.07044715122360769949189075127, −9.089576147155280763880666307735, −8.526720638005509941622774544361, −7.70740184129458314266730437975, −6.50972874191384263241051966681, −4.84476854274859383418376490847, −4.42685800476187952349813914739, −2.42657456451371677420339452810,
1.92021620124386544168703851796, 3.46442597693989909781215474088, 4.42744315082238295424344595261, 5.76732915102700888581277247625, 7.49004202768367172340437004625, 8.157020950839609029145760871878, 9.872910983680889185903720810725, 10.53179634832483546284077888940, 11.37505400016911591460110738012, 12.34520509094342812965642774733