| L(s) = 1 | + (−1.20 + 0.742i)2-s + (1.24 − 1.20i)3-s + (0.898 − 1.78i)4-s + (−3.07 − 1.11i)5-s + (−0.597 + 2.37i)6-s + (1.33 + 0.235i)7-s + (0.245 + 2.81i)8-s + (0.0801 − 2.99i)9-s + (4.53 − 0.935i)10-s + (−0.982 − 2.69i)11-s + (−1.04 − 3.30i)12-s + (−2.36 − 2.82i)13-s + (−1.78 + 0.709i)14-s + (−5.16 + 2.32i)15-s + (−2.38 − 3.20i)16-s + (1.94 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.851 + 0.524i)2-s + (0.716 − 0.697i)3-s + (0.449 − 0.893i)4-s + (−1.37 − 0.500i)5-s + (−0.243 + 0.969i)6-s + (0.505 + 0.0891i)7-s + (0.0867 + 0.996i)8-s + (0.0267 − 0.999i)9-s + (1.43 − 0.295i)10-s + (−0.296 − 0.813i)11-s + (−0.301 − 0.953i)12-s + (−0.656 − 0.782i)13-s + (−0.477 + 0.189i)14-s + (−1.33 + 0.600i)15-s + (−0.596 − 0.802i)16-s + (0.470 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0757 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0757 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.552328 - 0.511957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.552328 - 0.511957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.20 - 0.742i)T \) |
| 3 | \( 1 + (-1.24 + 1.20i)T \) |
| good | 5 | \( 1 + (3.07 + 1.11i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.33 - 0.235i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.982 + 2.69i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.36 + 2.82i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.22 - 2.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.08 + 6.17i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.00 - 4.19i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.65 + 0.820i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.70 + 2.14i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.28 - 8.67i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.24 - 1.17i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.21 - 12.5i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 + (-3.92 + 10.7i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-9.59 - 1.69i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.64 - 5.57i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.19 + 5.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.31 + 6.33i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.56 - 1.85i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.06 - 0.612i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.9 + 6.17i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.04692104876332291043023813747, −11.15832995853174574344553751184, −9.924805362876526239875436097528, −8.563880182915211623535307970901, −8.138637626318359169057906485508, −7.53761176117898789275158592257, −6.22069696891122306515284873409, −4.68985294893960001925860086686, −2.87375705112096463732880554806, −0.789709982169178284214100620463,
2.35465921742243513065280966660, 3.70298835406923989652387608868, 4.59096163079226827712479223348, 7.13015440157109285199448442814, 7.75139072465448924522613445671, 8.581039591994922101415023316898, 9.736894712285650610687584283910, 10.47285423139495437742228155642, 11.52858670102374363016861682926, 12.03119308985983005805843754126
