| L(s) = 1 | − 16·2-s + 256·4-s + 625·5-s + 4.65e3·7-s − 4.09e3·8-s − 1.00e4·10-s − 2.89e4·11-s − 1.64e5·13-s − 7.45e4·14-s + 6.55e4·16-s + 5.94e5·17-s − 2.95e5·19-s + 1.60e5·20-s + 4.63e5·22-s − 2.54e6·23-s + 3.90e5·25-s + 2.63e6·26-s + 1.19e6·28-s + 3.72e6·29-s + 2.33e6·31-s − 1.04e6·32-s − 9.51e6·34-s + 2.91e6·35-s + 1.08e7·37-s + 4.73e6·38-s − 2.56e6·40-s − 2.15e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.733·7-s − 0.353·8-s − 0.316·10-s − 0.597·11-s − 1.59·13-s − 0.518·14-s + 1/4·16-s + 1.72·17-s − 0.520·19-s + 0.223·20-s + 0.422·22-s − 1.89·23-s + 1/5·25-s + 1.12·26-s + 0.366·28-s + 0.977·29-s + 0.454·31-s − 0.176·32-s − 1.22·34-s + 0.327·35-s + 0.950·37-s + 0.368·38-s − 0.158·40-s − 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{4} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{4} T \) |
| good | 7 | \( 1 - 4658 T + p^{9} T^{2} \) |
| 11 | \( 1 + 28992 T + p^{9} T^{2} \) |
| 13 | \( 1 + 164446 T + p^{9} T^{2} \) |
| 17 | \( 1 - 594822 T + p^{9} T^{2} \) |
| 19 | \( 1 + 295780 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2544534 T + p^{9} T^{2} \) |
| 29 | \( 1 - 3722970 T + p^{9} T^{2} \) |
| 31 | \( 1 - 2335772 T + p^{9} T^{2} \) |
| 37 | \( 1 - 10840418 T + p^{9} T^{2} \) |
| 41 | \( 1 + 21593862 T + p^{9} T^{2} \) |
| 43 | \( 1 - 10832294 T + p^{9} T^{2} \) |
| 47 | \( 1 + 5172138 T + p^{9} T^{2} \) |
| 53 | \( 1 + 98179674 T + p^{9} T^{2} \) |
| 59 | \( 1 + 16162860 T + p^{9} T^{2} \) |
| 61 | \( 1 + 43928158 T + p^{9} T^{2} \) |
| 67 | \( 1 + 81557422 T + p^{9} T^{2} \) |
| 71 | \( 1 + 161307732 T + p^{9} T^{2} \) |
| 73 | \( 1 + 247147966 T + p^{9} T^{2} \) |
| 79 | \( 1 + 583345720 T + p^{9} T^{2} \) |
| 83 | \( 1 - 14571786 T + p^{9} T^{2} \) |
| 89 | \( 1 + 470133690 T + p^{9} T^{2} \) |
| 97 | \( 1 + 117838462 T + p^{9} T^{2} \) |
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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
−11.73751635273366729734367040201, −10.26893227557486934245910443548, −9.811206253901575220545187139856, −8.232248994847068397481908318324, −7.53856680048968592852546674258, −5.98804544176439375352420400606, −4.75615919298619735379031681385, −2.75262513499646953431111490281, −1.56800343779292777121343907912, 0,
1.56800343779292777121343907912, 2.75262513499646953431111490281, 4.75615919298619735379031681385, 5.98804544176439375352420400606, 7.53856680048968592852546674258, 8.232248994847068397481908318324, 9.811206253901575220545187139856, 10.26893227557486934245910443548, 11.73751635273366729734367040201
