| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s − 2.45e3·5-s + 1.29e3·6-s − 1.14e4·7-s − 4.09e3·8-s + 6.56e3·9-s + 3.92e4·10-s + 6.22e4·11-s − 2.07e4·12-s + 1.65e5·13-s + 1.82e5·14-s + 1.98e5·15-s + 6.55e4·16-s − 8.35e4·17-s − 1.04e5·18-s − 7.78e5·19-s − 6.28e5·20-s + 9.25e5·21-s − 9.96e5·22-s + 1.29e6·23-s + 3.31e5·24-s + 4.07e6·25-s − 2.64e6·26-s − 5.31e5·27-s − 2.92e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.75·5-s + 0.408·6-s − 1.79·7-s − 0.353·8-s + 0.333·9-s + 1.24·10-s + 1.28·11-s − 0.288·12-s + 1.60·13-s + 1.27·14-s + 1.01·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.878·20-s + 1.03·21-s − 0.907·22-s + 0.967·23-s + 0.204·24-s + 2.08·25-s − 1.13·26-s − 0.192·27-s − 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 16T \) |
| 3 | \( 1 + 81T \) |
| 17 | \( 1 + 8.35e4T \) |
| good | 5 | \( 1 + 2.45e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.14e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.22e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.65e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 7.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.29e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.94e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.19e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.89e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.36e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.48e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.50e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.64e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.29e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.52e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.79e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.36e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.07e8T + 7.60e17T^{2} \) |
| 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
−11.42188791322990230855586774165, −10.58653858894986869637864966444, −9.196396598252049435979873647852, −8.337799627954032389161844762638, −6.77717241702754733219902159105, −6.41313768170534137203718219760, −4.09404730306663039499949481892, −3.31980134273535418419239643092, −0.945738980675199760746925605121, 0,
0.945738980675199760746925605121, 3.31980134273535418419239643092, 4.09404730306663039499949481892, 6.41313768170534137203718219760, 6.77717241702754733219902159105, 8.337799627954032389161844762638, 9.196396598252049435979873647852, 10.58653858894986869637864966444, 11.42188791322990230855586774165
