L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 13-s − 2·15-s − 8·17-s + 6·19-s + 4·21-s + 6·23-s − 25-s − 27-s − 2·29-s + 2·31-s + 33-s − 8·35-s + 4·37-s + 39-s + 2·41-s − 4·43-s + 2·45-s − 4·47-s + 9·49-s + 8·51-s − 8·53-s − 2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s − 1.94·17-s + 1.37·19-s + 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.174·33-s − 1.35·35-s + 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.583·47-s + 9/7·49-s + 1.12·51-s − 1.09·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.106464907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.106464907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.77449283407253786009460886713, −7.06563821912503711347301271655, −6.37527899037507766078006035833, −6.10020985198044617350989038822, −5.14909446151884675735052829793, −4.59172296811147478255924579412, −3.42153359247955952503422970484, −2.77339832615760976506042506736, −1.83546111330411908461338448749, −0.53483098955708202235056274838,
0.53483098955708202235056274838, 1.83546111330411908461338448749, 2.77339832615760976506042506736, 3.42153359247955952503422970484, 4.59172296811147478255924579412, 5.14909446151884675735052829793, 6.10020985198044617350989038822, 6.37527899037507766078006035833, 7.06563821912503711347301271655, 7.77449283407253786009460886713