L(s) = 1 | + (−0.415 − 0.909i)2-s + (−2.45 + 0.721i)3-s + (−0.654 + 0.755i)4-s + (−3.26 − 2.10i)5-s + (1.67 + 1.93i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (2.99 − 1.92i)9-s + (−0.553 + 3.84i)10-s + (0.793 − 1.73i)11-s + (1.06 − 2.32i)12-s + (−0.713 + 4.96i)13-s + (−0.841 + 0.540i)14-s + (9.54 + 2.80i)15-s + (−0.142 − 0.989i)16-s + (4.11 + 4.75i)17-s + ⋯ |
L(s) = 1 | + (−0.293 − 0.643i)2-s + (−1.41 + 0.416i)3-s + (−0.327 + 0.377i)4-s + (−1.46 − 0.939i)5-s + (0.684 + 0.789i)6-s + (−0.0537 − 0.374i)7-s + (0.339 + 0.0996i)8-s + (0.997 − 0.640i)9-s + (−0.174 + 1.21i)10-s + (0.239 − 0.523i)11-s + (0.307 − 0.672i)12-s + (−0.197 + 1.37i)13-s + (−0.224 + 0.144i)14-s + (2.46 + 0.723i)15-s + (−0.0355 − 0.247i)16-s + (0.998 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.344785 + 0.0887309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.344785 + 0.0887309i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.14 + 2.41i)T \) |
good | 3 | \( 1 + (2.45 - 0.721i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (3.26 + 2.10i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (-0.793 + 1.73i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.713 - 4.96i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.11 - 4.75i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.84 - 2.13i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.38 + 1.60i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (2.75 + 0.808i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.51 + 0.976i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (0.951 + 0.611i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (7.22 - 2.12i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 + (-1.97 - 13.7i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 9.45i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-10.4 - 3.07i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-4.77 - 10.4i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.74 + 3.82i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.24 - 1.43i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.80 - 12.5i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (8.38 - 5.38i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (14.8 - 4.35i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-5.45 - 3.50i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.55217902797386672592742208654, −11.07985103720392274657483475526, −10.12497888020610465260878481144, −8.948745322107305497722231683952, −8.083721906892603555690989040386, −6.88099223533847872283112300149, −5.53038682750839257106651814710, −4.35110419313751614152307958134, −3.85751712065039972810756084947, −1.04291172126026529596400349932,
0.44368005304758956518157148198, 3.24084945108201283662138765437, 4.87327628819476268778505592214, 5.71388689950664445789635610700, 7.11899325742899319658833039391, 7.17202481072277690673329883646, 8.382249955087763578949565726747, 9.900015479165536672783500979540, 10.82268140684096871185828694637, 11.53391640114938052197648201483