| L(s) = 1 | + 4·2-s + 16·4-s + 84·5-s + 64·8-s + 336·10-s + 336·11-s − 584·13-s + 256·16-s − 1.45e3·17-s − 470·19-s + 1.34e3·20-s + 1.34e3·22-s + 4.20e3·23-s + 3.93e3·25-s − 2.33e3·26-s − 4.86e3·29-s + 7.37e3·31-s + 1.02e3·32-s − 5.83e3·34-s + 1.43e4·37-s − 1.88e3·38-s + 5.37e3·40-s + 6.22e3·41-s + 3.70e3·43-s + 5.37e3·44-s + 1.68e4·46-s − 1.81e3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.50·5-s + 0.353·8-s + 1.06·10-s + 0.837·11-s − 0.958·13-s + 1/4·16-s − 1.22·17-s − 0.298·19-s + 0.751·20-s + 0.592·22-s + 1.65·23-s + 1.25·25-s − 0.677·26-s − 1.07·29-s + 1.37·31-s + 0.176·32-s − 0.865·34-s + 1.72·37-s − 0.211·38-s + 0.531·40-s + 0.578·41-s + 0.305·43-s + 0.418·44-s + 1.17·46-s − 0.119·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.415649272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.415649272\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 84 T + p^{5} T^{2} \) |
| 11 | \( 1 - 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 584 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 + 470 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 + 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 + 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 - 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 - 16978 T + p^{5} T^{2} \) |
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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
−9.408973731031721803087585351830, −8.823972315927494206927386426557, −7.36977669472069384855626918636, −6.59045773130498343541670156850, −5.94016220456823706351040465501, −4.99888732023323656481769089754, −4.21626089454046941455525681040, −2.75217827729155760703739467509, −2.12637839834998106646471469788, −0.964197451088683002451190468064,
0.964197451088683002451190468064, 2.12637839834998106646471469788, 2.75217827729155760703739467509, 4.21626089454046941455525681040, 4.99888732023323656481769089754, 5.94016220456823706351040465501, 6.59045773130498343541670156850, 7.36977669472069384855626918636, 8.823972315927494206927386426557, 9.408973731031721803087585351830
