| L(s) = 1 | + 18·2-s + 81·3-s − 188·4-s − 1.53e3·5-s + 1.45e3·6-s + 9.12e3·7-s − 1.26e4·8-s + 6.56e3·9-s − 2.75e4·10-s + 2.11e4·11-s − 1.52e4·12-s + 3.12e4·13-s + 1.64e5·14-s − 1.23e5·15-s − 1.30e5·16-s − 2.79e5·17-s + 1.18e5·18-s + 1.44e5·19-s + 2.87e5·20-s + 7.39e5·21-s + 3.80e5·22-s − 1.76e6·23-s − 1.02e6·24-s + 3.87e5·25-s + 5.61e5·26-s + 5.31e5·27-s − 1.71e6·28-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 0.577·3-s − 0.367·4-s − 1.09·5-s + 0.459·6-s + 1.43·7-s − 1.08·8-s + 1/3·9-s − 0.870·10-s + 0.435·11-s − 0.211·12-s + 0.303·13-s + 1.14·14-s − 0.632·15-s − 0.497·16-s − 0.811·17-s + 0.265·18-s + 0.253·19-s + 0.401·20-s + 0.829·21-s + 0.346·22-s − 1.31·23-s − 0.627·24-s + 0.198·25-s + 0.241·26-s + 0.192·27-s − 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.582503348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582503348\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 - 9 p T + p^{9} T^{2} \) |
| 5 | \( 1 + 306 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 1304 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 21132 T + p^{9} T^{2} \) |
| 13 | \( 1 - 31214 T + p^{9} T^{2} \) |
| 17 | \( 1 + 279342 T + p^{9} T^{2} \) |
| 19 | \( 1 - 7580 p T + p^{9} T^{2} \) |
| 23 | \( 1 + 1763496 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4692510 T + p^{9} T^{2} \) |
| 31 | \( 1 + 369088 T + p^{9} T^{2} \) |
| 37 | \( 1 - 9347078 T + p^{9} T^{2} \) |
| 41 | \( 1 + 7226838 T + p^{9} T^{2} \) |
| 43 | \( 1 + 23147476 T + p^{9} T^{2} \) |
| 47 | \( 1 - 22971888 T + p^{9} T^{2} \) |
| 53 | \( 1 - 78477174 T + p^{9} T^{2} \) |
| 59 | \( 1 + 20310660 T + p^{9} T^{2} \) |
| 61 | \( 1 + 179339938 T + p^{9} T^{2} \) |
| 67 | \( 1 - 274528388 T + p^{9} T^{2} \) |
| 71 | \( 1 + 36342648 T + p^{9} T^{2} \) |
| 73 | \( 1 + 247089526 T + p^{9} T^{2} \) |
| 79 | \( 1 - 191874800 T + p^{9} T^{2} \) |
| 83 | \( 1 + 276159276 T + p^{9} T^{2} \) |
| 89 | \( 1 + 678997350 T + p^{9} T^{2} \) |
| 97 | \( 1 + 567657502 T + p^{9} T^{2} \) |
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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
−24.45634779544886684522432975618, −23.42428523592690378448300201885, −21.67697623036061767097066958924, −20.07847850703538068957892920589, −18.11149703708407986621089400968, −15.29624656455437783716188521345, −13.98380752793585255956543148336, −11.80494425077708071454074581219, −8.330166823448443637878380766491, −4.26750581310523036110086468336,
4.26750581310523036110086468336, 8.330166823448443637878380766491, 11.80494425077708071454074581219, 13.98380752793585255956543148336, 15.29624656455437783716188521345, 18.11149703708407986621089400968, 20.07847850703538068957892920589, 21.67697623036061767097066958924, 23.42428523592690378448300201885, 24.45634779544886684522432975618
