L(s) = 1 | + 2.92·2-s − 9·3-s − 23.4·4-s − 26.2·6-s + 85.0·7-s − 162.·8-s + 81·9-s − 121·11-s + 211.·12-s − 724.·13-s + 248.·14-s + 277.·16-s + 2.09e3·17-s + 236.·18-s − 6.40·19-s − 765.·21-s − 353.·22-s − 1.56e3·23-s + 1.45e3·24-s − 2.11e3·26-s − 729·27-s − 1.99e3·28-s − 5.14e3·29-s − 1.03e3·31-s + 5.99e3·32-s + 1.08e3·33-s + 6.13e3·34-s + ⋯ |
L(s) = 1 | + 0.516·2-s − 0.577·3-s − 0.733·4-s − 0.298·6-s + 0.655·7-s − 0.895·8-s + 0.333·9-s − 0.301·11-s + 0.423·12-s − 1.18·13-s + 0.338·14-s + 0.270·16-s + 1.76·17-s + 0.172·18-s − 0.00406·19-s − 0.378·21-s − 0.155·22-s − 0.618·23-s + 0.516·24-s − 0.613·26-s − 0.192·27-s − 0.480·28-s − 1.13·29-s − 0.192·31-s + 1.03·32-s + 0.174·33-s + 0.909·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 2.92T + 32T^{2} \) |
| 7 | \( 1 - 85.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 724.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 6.40T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
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
−9.294962060099826244160204489984, −7.914330285723199381888454815096, −7.55712335077672776302365328732, −6.02916301198691018755960297134, −5.42747433102649607331012164044, −4.68436205643931953604344196178, −3.81257051325707501448787147813, −2.56557075570168019424297531847, −1.10079465515973933975113574052, 0,
1.10079465515973933975113574052, 2.56557075570168019424297531847, 3.81257051325707501448787147813, 4.68436205643931953604344196178, 5.42747433102649607331012164044, 6.02916301198691018755960297134, 7.55712335077672776302365328732, 7.914330285723199381888454815096, 9.294962060099826244160204489984