L(s) = 1 | + (1.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + (2 − 3.46i)13-s + (−1.5 + 0.866i)15-s − 6·17-s + 2·19-s + 1.73i·21-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + 5.19i·27-s + (−1.5 − 2.59i)29-s + (5 − 8.66i)31-s − 0.999·35-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + (0.5 + 0.866i)9-s + (0.554 − 0.960i)13-s + (−0.387 + 0.223i)15-s − 1.45·17-s + 0.458·19-s + 0.377i·21-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + 0.999i·27-s + (−0.278 − 0.482i)29-s + (0.898 − 1.55i)31-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37632 + 0.500939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37632 + 0.500939i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.07323100536743089832902151211, −11.61554228388570923308019590650, −10.71872216541710044944359507107, −9.764251191120569755496157805707, −8.637834542706514113266809170384, −7.923449841738607293638115484093, −6.57801519763663424795933877212, −5.02867996757359366970880323523, −3.72024618991411865920399568644, −2.43901192103932383202408943676,
1.67886815582823748703660128397, 3.46873584516959243507074780252, 4.71978973954967404955287113488, 6.54779774330716892423916671853, 7.39525228783872442131807080311, 8.669759085305776939208420995501, 9.131490110610312350342747870835, 10.59624379239894998133655309897, 11.72487625931839831993825130033, 12.64695803371730144828687322633