L(s) = 1 | − 1.73·5-s − 2·7-s + 3.46·11-s + 13-s − 5.19·17-s + 2·19-s + 3.46·23-s − 2.00·25-s + 1.73·29-s − 8·31-s + 3.46·35-s + 7·37-s + 6.92·41-s + 2·43-s + 6.92·47-s − 3·49-s − 5.99·55-s − 13.8·59-s + 7·61-s − 1.73·65-s − 10·67-s − 10.3·71-s − 7·73-s − 6.92·77-s − 2·79-s + 13.8·83-s + 9·85-s + ⋯ |
L(s) = 1 | − 0.774·5-s − 0.755·7-s + 1.04·11-s + 0.277·13-s − 1.26·17-s + 0.458·19-s + 0.722·23-s − 0.400·25-s + 0.321·29-s − 1.43·31-s + 0.585·35-s + 1.15·37-s + 1.08·41-s + 0.304·43-s + 1.01·47-s − 0.428·49-s − 0.809·55-s − 1.80·59-s + 0.896·61-s − 0.214·65-s − 1.22·67-s − 1.23·71-s − 0.819·73-s − 0.789·77-s − 0.225·79-s + 1.52·83-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.66118358993559354695247348624, −7.23957013552453805696358320914, −6.37961363568063911587678406325, −5.90854682240406164865750131292, −4.71294511783717167141647595679, −4.05632433172871795397603711677, −3.43169400473798997744102011524, −2.49026189652748061274839834559, −1.23245567438395537302402289373, 0,
1.23245567438395537302402289373, 2.49026189652748061274839834559, 3.43169400473798997744102011524, 4.05632433172871795397603711677, 4.71294511783717167141647595679, 5.90854682240406164865750131292, 6.37961363568063911587678406325, 7.23957013552453805696358320914, 7.66118358993559354695247348624