| L(s) = 1 | − 28·2-s + 116·3-s + 272·4-s − 3.24e3·6-s − 2.40e3·7-s + 6.72e3·8-s − 6.22e3·9-s − 2.55e4·11-s + 3.15e4·12-s + 4.23e4·13-s + 6.72e4·14-s − 3.27e5·16-s + 5.26e5·17-s + 1.74e5·18-s − 3.50e5·19-s − 2.78e5·21-s + 7.15e5·22-s + 6.21e5·23-s + 7.79e5·24-s − 1.18e6·26-s − 3.00e6·27-s − 6.53e5·28-s + 6.72e6·29-s − 6.41e6·31-s + 5.72e6·32-s − 2.96e6·33-s − 1.47e7·34-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.826·3-s + 0.531·4-s − 1.02·6-s − 0.377·7-s + 0.580·8-s − 0.316·9-s − 0.526·11-s + 0.439·12-s + 0.410·13-s + 0.467·14-s − 1.24·16-s + 1.52·17-s + 0.391·18-s − 0.616·19-s − 0.312·21-s + 0.651·22-s + 0.463·23-s + 0.479·24-s − 0.508·26-s − 1.08·27-s − 0.200·28-s + 1.76·29-s − 1.24·31-s + 0.965·32-s − 0.435·33-s − 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + p^{4} T \) |
| good | 2 | \( 1 + 7 p^{2} T + p^{9} T^{2} \) |
| 3 | \( 1 - 116 T + p^{9} T^{2} \) |
| 11 | \( 1 + 25548 T + p^{9} T^{2} \) |
| 13 | \( 1 - 42306 T + p^{9} T^{2} \) |
| 17 | \( 1 - 526342 T + p^{9} T^{2} \) |
| 19 | \( 1 + 350060 T + p^{9} T^{2} \) |
| 23 | \( 1 - 621976 T + p^{9} T^{2} \) |
| 29 | \( 1 - 6720430 T + p^{9} T^{2} \) |
| 31 | \( 1 + 6412208 T + p^{9} T^{2} \) |
| 37 | \( 1 - 2317682 T + p^{9} T^{2} \) |
| 41 | \( 1 + 10224678 T + p^{9} T^{2} \) |
| 43 | \( 1 + 30114004 T + p^{9} T^{2} \) |
| 47 | \( 1 - 23644912 T + p^{9} T^{2} \) |
| 53 | \( 1 + 57292654 T + p^{9} T^{2} \) |
| 59 | \( 1 - 84934780 T + p^{9} T^{2} \) |
| 61 | \( 1 - 14677822 T + p^{9} T^{2} \) |
| 67 | \( 1 - 244557812 T + p^{9} T^{2} \) |
| 71 | \( 1 - 61901952 T + p^{9} T^{2} \) |
| 73 | \( 1 - 283763726 T + p^{9} T^{2} \) |
| 79 | \( 1 - 276107480 T + p^{9} T^{2} \) |
| 83 | \( 1 - 72995956 T + p^{9} T^{2} \) |
| 89 | \( 1 + 896368470 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1205809578 T + p^{9} T^{2} \) |
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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
−10.19112103600048002666938633816, −9.435712170282542004710924262217, −8.439744059232341352264003846412, −7.977719110662855327091087384402, −6.76732199824049326380432917954, −5.27286289517867541834394924791, −3.63840962846763431560313798285, −2.52153495373356647184053484745, −1.21349497488542371736328244652, 0,
1.21349497488542371736328244652, 2.52153495373356647184053484745, 3.63840962846763431560313798285, 5.27286289517867541834394924791, 6.76732199824049326380432917954, 7.977719110662855327091087384402, 8.439744059232341352264003846412, 9.435712170282542004710924262217, 10.19112103600048002666938633816
