| L(s) = 1 | + 81·3-s + 794·5-s + 5.88e3·7-s + 6.56e3·9-s − 3.06e4·11-s + 1.53e4·13-s + 6.43e4·15-s − 5.75e5·17-s − 6.17e5·19-s + 4.76e5·21-s − 4.41e5·23-s − 1.32e6·25-s + 5.31e5·27-s + 2.32e6·29-s − 9.58e6·31-s − 2.48e6·33-s + 4.66e6·35-s − 9.27e6·37-s + 1.24e6·39-s − 5.90e6·41-s + 3.35e7·43-s + 5.20e6·45-s − 2.11e7·47-s − 5.77e6·49-s − 4.65e7·51-s + 1.08e8·53-s − 2.43e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.568·5-s + 0.925·7-s + 1/3·9-s − 0.631·11-s + 0.148·13-s + 0.328·15-s − 1.66·17-s − 1.08·19-s + 0.534·21-s − 0.329·23-s − 0.677·25-s + 0.192·27-s + 0.611·29-s − 1.86·31-s − 0.364·33-s + 0.525·35-s − 0.813·37-s + 0.0858·39-s − 0.326·41-s + 1.49·43-s + 0.189·45-s − 0.631·47-s − 0.143·49-s − 0.964·51-s + 1.89·53-s − 0.358·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| good | 5 | \( 1 - 794 T + p^{9} T^{2} \) |
| 7 | \( 1 - 120 p^{2} T + p^{9} T^{2} \) |
| 11 | \( 1 + 30644 T + p^{9} T^{2} \) |
| 13 | \( 1 - 1178 p T + p^{9} T^{2} \) |
| 17 | \( 1 + 575086 T + p^{9} T^{2} \) |
| 19 | \( 1 + 617644 T + p^{9} T^{2} \) |
| 23 | \( 1 + 441880 T + p^{9} T^{2} \) |
| 29 | \( 1 - 80298 p T + p^{9} T^{2} \) |
| 31 | \( 1 + 9588512 T + p^{9} T^{2} \) |
| 37 | \( 1 + 9276678 T + p^{9} T^{2} \) |
| 41 | \( 1 + 5903766 T + p^{9} T^{2} \) |
| 43 | \( 1 - 33593452 T + p^{9} T^{2} \) |
| 47 | \( 1 + 21135408 T + p^{9} T^{2} \) |
| 53 | \( 1 - 108575594 T + p^{9} T^{2} \) |
| 59 | \( 1 + 127636868 T + p^{9} T^{2} \) |
| 61 | \( 1 + 147189214 T + p^{9} T^{2} \) |
| 67 | \( 1 + 33157756 T + p^{9} T^{2} \) |
| 71 | \( 1 - 9293752 T + p^{9} T^{2} \) |
| 73 | \( 1 - 351080074 T + p^{9} T^{2} \) |
| 79 | \( 1 - 126193328 T + p^{9} T^{2} \) |
| 83 | \( 1 - 475037588 T + p^{9} T^{2} \) |
| 89 | \( 1 + 566133990 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1474684318 T + p^{9} T^{2} \) |
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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
−10.49033396611517647056463452206, −9.203860852741009887551897011410, −8.473081816909170079248364181301, −7.46573317894391538681755301662, −6.25110312277708882107751794772, −5.01548986897676732425990160395, −3.98258481812191098721873462647, −2.37583879494105906239233737135, −1.74200310007033404424858643147, 0,
1.74200310007033404424858643147, 2.37583879494105906239233737135, 3.98258481812191098721873462647, 5.01548986897676732425990160395, 6.25110312277708882107751794772, 7.46573317894391538681755301662, 8.473081816909170079248364181301, 9.203860852741009887551897011410, 10.49033396611517647056463452206
