L(s) = 1 | + (0.0864 + 0.149i)2-s + (0.985 − 1.70i)4-s + (−1.86 + 3.23i)5-s + (−1.51 − 2.62i)7-s + 0.686·8-s − 0.646·10-s + (−1.24 − 2.15i)11-s + (0.382 − 0.662i)13-s + (0.262 − 0.454i)14-s + (−1.91 − 3.30i)16-s − 4.62·17-s − 0.611·19-s + (3.68 + 6.37i)20-s + (0.215 − 0.373i)22-s + (3.26 − 5.65i)23-s + ⋯ |
L(s) = 1 | + (0.0611 + 0.105i)2-s + (0.492 − 0.853i)4-s + (−0.835 + 1.44i)5-s + (−0.572 − 0.992i)7-s + 0.242·8-s − 0.204·10-s + (−0.375 − 0.650i)11-s + (0.106 − 0.183i)13-s + (0.0700 − 0.121i)14-s + (−0.477 − 0.827i)16-s − 1.12·17-s − 0.140·19-s + (0.823 + 1.42i)20-s + (0.0459 − 0.0795i)22-s + (0.680 − 1.17i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.389061 - 0.673875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.389061 - 0.673875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.0864 - 0.149i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.86 - 3.23i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.51 + 2.62i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.24 + 2.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.382 + 0.662i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 + 0.611T + 19T^{2} \) |
| 23 | \( 1 + (-3.26 + 5.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.27 + 5.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.27 - 5.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 + (-2.63 + 4.55i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.78 + 4.82i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.553 - 0.959i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 + (5.92 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.09 + 7.09i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.606 + 1.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + (-5.89 - 10.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.50 - 7.80i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 + (-0.474 - 0.821i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.52101974596348809731863288497, −9.477958266585090248851411551734, −8.166973954349805366076269801237, −7.14534917594546971262687262458, −6.76088714147951488576534681720, −5.94058423410793550206935583879, −4.48872805422083980398877964233, −3.43499151416229099517119035611, −2.45236387474715049476445580038, −0.36808030650354557377827098369,
1.87330447859963838101780506244, 3.17195365667453867357634180174, 4.26167788645797793677730789379, 5.08652586481928254245805698958, 6.29744007659574312309626431420, 7.44495892294792785942431371486, 8.071181574314172379531402590266, 9.063554679754861369093948665336, 9.365540650670453256477851306169, 11.07076021668948645169545005284