| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s + 2.25e3·5-s + 1.29e3·6-s + 7.81e3·7-s − 4.09e3·8-s + 6.56e3·9-s − 3.61e4·10-s − 8.82e4·11-s − 2.07e4·12-s − 5.83e4·13-s − 1.25e5·14-s − 1.82e5·15-s + 6.55e4·16-s − 8.35e4·17-s − 1.04e5·18-s − 4.69e5·19-s + 5.77e5·20-s − 6.32e5·21-s + 1.41e6·22-s + 7.54e5·23-s + 3.31e5·24-s + 3.14e6·25-s + 9.34e5·26-s − 5.31e5·27-s + 2.00e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.61·5-s + 0.408·6-s + 1.23·7-s − 0.353·8-s + 0.333·9-s − 1.14·10-s − 1.81·11-s − 0.288·12-s − 0.566·13-s − 0.869·14-s − 0.932·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.826·19-s + 0.807·20-s − 0.710·21-s + 1.28·22-s + 0.562·23-s + 0.204·24-s + 1.60·25-s + 0.400·26-s − 0.192·27-s + 0.615·28-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 | \( 1 - 2.25e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.81e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.83e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.69e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.54e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.35e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.83e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.87e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.73e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.82e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.67e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.90e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.26e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.62e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.66e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.59e9T + 7.60e17T^{2} \) |
| 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
−11.00177338052688712099525448259, −10.52996764817592260235496846668, −9.487944739303609915043836391679, −8.250275838249398113155912208417, −7.07868877396585534919791095978, −5.64250935847980790234855822434, −5.01864273838337973347320873361, −2.40026326711956703370351588262, −1.63307185839631150011849960957, 0,
1.63307185839631150011849960957, 2.40026326711956703370351588262, 5.01864273838337973347320873361, 5.64250935847980790234855822434, 7.07868877396585534919791095978, 8.250275838249398113155912208417, 9.487944739303609915043836391679, 10.52996764817592260235496846668, 11.00177338052688712099525448259
