L(s) = 1 | + 1.90·5-s + (−2.43 + 1.03i)7-s + 3.06·11-s + (1.13 − 1.96i)13-s + (0.713 − 1.23i)17-s + (2.98 + 5.16i)19-s + 7.15·23-s − 1.37·25-s + (−0.468 − 0.810i)29-s + (4.11 + 7.11i)31-s + (−4.63 + 1.97i)35-s + (−1.41 − 2.45i)37-s + (5.31 − 9.20i)41-s + (2.98 + 5.16i)43-s + (−0.483 + 0.837i)47-s + ⋯ |
L(s) = 1 | + 0.851·5-s + (−0.920 + 0.391i)7-s + 0.924·11-s + (0.313 − 0.543i)13-s + (0.173 − 0.299i)17-s + (0.684 + 1.18i)19-s + 1.49·23-s − 0.275·25-s + (−0.0869 − 0.150i)29-s + (0.738 + 1.27i)31-s + (−0.782 + 0.333i)35-s + (−0.232 − 0.403i)37-s + (0.830 − 1.43i)41-s + (0.455 + 0.788i)43-s + (−0.0705 + 0.122i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73783 + 0.206756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73783 + 0.206756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.43 - 1.03i)T \) |
good | 5 | \( 1 - 1.90T + 5T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + (-1.13 + 1.96i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.713 + 1.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.98 - 5.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 + (0.468 + 0.810i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.11 - 7.11i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.41 + 2.45i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.31 + 9.20i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.98 - 5.16i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.483 - 0.837i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.45 - 9.44i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.68 + 9.84i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.449 - 0.778i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.813 + 1.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 + (0.996 - 1.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.16 + 7.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.98 - 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.58 - 4.48i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.922 - 1.59i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.24083937624759235247026465838, −9.429714640560890457646119251639, −9.001348427173066376290815304846, −7.76654457607722545228649467600, −6.68878827969493266708429020137, −5.99283922778990466824216122433, −5.20594630303706090913277689758, −3.71772242504210065178842598563, −2.79724931865543122832227970783, −1.29676658614499144033608129306,
1.13775168616962812750866719739, 2.66046479425599718550152643975, 3.76863678775406306213962442683, 4.89938006564437742721685944840, 6.13405184965706786118258766777, 6.62122482147832434834557797246, 7.56978644817600626256805904219, 9.007001426757031445594979988424, 9.373119688833455535852552761427, 10.14213576416005893322176770369