L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 16.7·5-s − 6·6-s + 24.0·7-s + 8·8-s + 9·9-s − 33.4·10-s + 14.3·11-s − 12·12-s − 53.8·13-s + 48.0·14-s + 50.2·15-s + 16·16-s + 61.0·17-s + 18·18-s − 115.·19-s − 66.9·20-s − 72.0·21-s + 28.6·22-s − 188.·23-s − 24·24-s + 155.·25-s − 107.·26-s − 27·27-s + 96.0·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.49·5-s − 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s − 1.05·10-s + 0.392·11-s − 0.288·12-s − 1.14·13-s + 0.917·14-s + 0.864·15-s + 0.250·16-s + 0.871·17-s + 0.235·18-s − 1.39·19-s − 0.749·20-s − 0.748·21-s + 0.277·22-s − 1.70·23-s − 0.204·24-s + 1.24·25-s − 0.811·26-s − 0.192·27-s + 0.648·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
good | 5 | \( 1 + 16.7T + 125T^{2} \) |
| 7 | \( 1 - 24.0T + 343T^{2} \) |
| 11 | \( 1 - 14.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 22.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 380.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 18.7T + 1.48e5T^{2} \) |
| 61 | \( 1 - 719.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 365.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 341.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 226.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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
−11.00701986886811573549303841679, −9.954963867444308992168812416236, −8.233703924120726710124410060816, −7.76094783895474575534563955788, −6.73192157854847402695276045614, −5.37757823123400767544002951168, −4.49406296019897794126731729454, −3.74806354623674706393129482730, −1.88179002391441450122947694692, 0,
1.88179002391441450122947694692, 3.74806354623674706393129482730, 4.49406296019897794126731729454, 5.37757823123400767544002951168, 6.73192157854847402695276045614, 7.76094783895474575534563955788, 8.233703924120726710124410060816, 9.954963867444308992168812416236, 11.00701986886811573549303841679