L(s) = 1 | − 2-s − 4-s + (1.5 − 2.59i)5-s + (2 + 1.73i)7-s + 3·8-s + (−1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (−1 + 3.46i)13-s + (−2 − 1.73i)14-s − 16-s + 2·17-s + (0.5 + 0.866i)19-s + (−1.5 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−2 − 3.46i)25-s + (1 − 3.46i)26-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.5·4-s + (0.670 − 1.16i)5-s + (0.755 + 0.654i)7-s + 1.06·8-s + (−0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + (−0.277 + 0.960i)13-s + (−0.534 − 0.462i)14-s − 0.250·16-s + 0.485·17-s + (0.114 + 0.198i)19-s + (−0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.400 − 0.692i)25-s + (0.196 − 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06401 + 0.206852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06401 + 0.206852i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.939165597226645645218230901820, −9.343161768955426914645143453045, −8.746883799490829594903641028401, −8.044422464239794806373274107640, −7.10504011516982068271075081864, −5.58625417441522171535448808730, −4.99660545715066328729164023005, −4.23956269856436241918243244399, −2.18038063981077940198230753869, −1.23821931987795033679240901262,
0.835655756923601098163415042913, 2.43760343044770223921198018033, 3.62128743612012108477951849056, 4.90940484260439897130647317915, 5.79822073224444378540588248686, 6.94443125250656373439691471542, 7.85170734237166521008911248876, 8.300126757796985391135939740893, 9.561407804816850601755642414400, 10.19498515193657594743587468463