| L(s) = 1 | + 1.71·2-s − 1.04·3-s + 0.926·4-s − 4.04·5-s − 1.78·6-s − 1.83·8-s − 1.91·9-s − 6.91·10-s − 2.15·11-s − 0.964·12-s + 4.46·13-s + 4.20·15-s − 4.99·16-s − 1.91·17-s − 3.28·18-s + 0.111·19-s − 3.74·20-s − 3.68·22-s + 1.02·23-s + 1.91·24-s + 11.3·25-s + 7.63·26-s + 5.11·27-s − 5.13·29-s + 7.19·30-s − 6.70·31-s − 4.87·32-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.600·3-s + 0.463·4-s − 1.80·5-s − 0.726·6-s − 0.649·8-s − 0.639·9-s − 2.18·10-s − 0.648·11-s − 0.278·12-s + 1.23·13-s + 1.08·15-s − 1.24·16-s − 0.464·17-s − 0.773·18-s + 0.0255·19-s − 0.837·20-s − 0.785·22-s + 0.214·23-s + 0.389·24-s + 2.26·25-s + 1.49·26-s + 0.984·27-s − 0.952·29-s + 1.31·30-s − 1.20·31-s − 0.861·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 0.111T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 + 7.88T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 0.352T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 2.02T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.36659027066617885747335407449, −10.79860400287109806172755409957, −8.926579505867914071381024639837, −8.171356437489448767299763594488, −6.99506914824956191562136131722, −5.88909197976391247822617533925, −4.94549904174940187702144358118, −3.94509386590163929261692452859, −3.13782709236535073239015062810, 0,
3.13782709236535073239015062810, 3.94509386590163929261692452859, 4.94549904174940187702144358118, 5.88909197976391247822617533925, 6.99506914824956191562136131722, 8.171356437489448767299763594488, 8.926579505867914071381024639837, 10.79860400287109806172755409957, 11.36659027066617885747335407449
