L(s) = 1 | + 0.218·2-s − 3·3-s − 7.95·4-s + 5·5-s − 0.656·6-s − 20.9·7-s − 3.48·8-s + 9·9-s + 1.09·10-s − 1.46·11-s + 23.8·12-s − 79.4·13-s − 4.59·14-s − 15·15-s + 62.8·16-s − 65.4·17-s + 1.96·18-s + 116.·19-s − 39.7·20-s + 62.9·21-s − 0.320·22-s − 177.·23-s + 10.4·24-s + 25·25-s − 17.3·26-s − 27·27-s + 166.·28-s + ⋯ |
L(s) = 1 | + 0.0773·2-s − 0.577·3-s − 0.994·4-s + 0.447·5-s − 0.0446·6-s − 1.13·7-s − 0.154·8-s + 0.333·9-s + 0.0345·10-s − 0.0401·11-s + 0.573·12-s − 1.69·13-s − 0.0876·14-s − 0.258·15-s + 0.982·16-s − 0.933·17-s + 0.0257·18-s + 1.41·19-s − 0.444·20-s + 0.654·21-s − 0.00310·22-s − 1.60·23-s + 0.0890·24-s + 0.200·25-s − 0.131·26-s − 0.192·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7096604796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7096604796\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
good | 2 | \( 1 - 0.218T + 8T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 + 1.46T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 77.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 62.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 66.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 426.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 79.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 218.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 884.T + 9.12e5T^{2} \) |
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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
−10.42745597050537262801924540209, −9.692316285197396707153188638413, −9.357731139814288913072143888818, −7.936837887501491501547013412376, −6.87289691323759682479755572143, −5.85234417985659007857250130803, −5.00982384976872826631653443736, −3.96020834428211944202156491478, −2.54374036600031587190696941934, −0.53059391121560305721812868442,
0.53059391121560305721812868442, 2.54374036600031587190696941934, 3.96020834428211944202156491478, 5.00982384976872826631653443736, 5.85234417985659007857250130803, 6.87289691323759682479755572143, 7.936837887501491501547013412376, 9.357731139814288913072143888818, 9.692316285197396707153188638413, 10.42745597050537262801924540209