L(s) = 1 | + (−0.866 − 0.5i)3-s + i·5-s + (−0.5 + 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)11-s + (0.5 − 0.866i)15-s + (0.866 − 0.499i)21-s − 25-s − 0.999i·27-s − 1.73·29-s + (0.5 − 0.866i)31-s + (1.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.499i)45-s + (−0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + i·5-s + (−0.5 + 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)11-s + (0.5 − 0.866i)15-s + (0.866 − 0.499i)21-s − 25-s − 0.999i·27-s − 1.73·29-s + (0.5 − 0.866i)31-s + (1.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.499i)45-s + (−0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3444219365\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3444219365\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 11 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−9.461932143595083949931809500532, −8.117544928299998900582394344207, −7.51996803469776318838975272327, −6.89444322985583510334351638371, −6.22034492476518892717897659256, −5.51202500529077856338661037780, −4.79466723302134564493009361319, −3.66137186410304829870240862333, −2.44973243318153198766099807534, −1.98132035817400889090460418254,
0.23366001741253895896962949046, 1.30573304650371896594606997984, 3.07041191798465419482441483614, 3.87305855505506616878973324106, 4.61284434646938418228667398628, 5.53229203373696301998289634732, 5.86961653013272374268652792076, 6.87375195616329808010894907652, 7.70517662559185341051148732955, 8.522609627768367972869877201521