| L(s) = 1 | − 0.229·2-s − 7.94·4-s − 5·5-s + 7·7-s + 3.66·8-s + 1.14·10-s + 51.0·11-s + 46.0·13-s − 1.60·14-s + 62.7·16-s − 72.7·17-s − 123.·19-s + 39.7·20-s − 11.7·22-s − 156.·23-s + 25·25-s − 10.5·26-s − 55.6·28-s − 191.·29-s − 116.·31-s − 43.7·32-s + 16.7·34-s − 35·35-s + 83.1·37-s + 28.4·38-s − 18.3·40-s − 466.·41-s + ⋯ |
| L(s) = 1 | − 0.0812·2-s − 0.993·4-s − 0.447·5-s + 0.377·7-s + 0.161·8-s + 0.0363·10-s + 1.39·11-s + 0.983·13-s − 0.0307·14-s + 0.980·16-s − 1.03·17-s − 1.49·19-s + 0.444·20-s − 0.113·22-s − 1.41·23-s + 0.200·25-s − 0.0798·26-s − 0.375·28-s − 1.22·29-s − 0.673·31-s − 0.241·32-s + 0.0843·34-s − 0.169·35-s + 0.369·37-s + 0.121·38-s − 0.0724·40-s − 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 0.229T + 8T^{2} \) |
| 11 | \( 1 - 51.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 114.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 214.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 366.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.06e3T + 9.12e5T^{2} \) |
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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
−10.86346312755941098979034917844, −9.663395962957927789884371756961, −8.724989337689837375123592810194, −8.259430965560902117938637901779, −6.83508950229635540164151879136, −5.77795834298411916829247256156, −4.26248153342305032211607885846, −3.87528630901471435766539165442, −1.66079032692313443558990512394, 0,
1.66079032692313443558990512394, 3.87528630901471435766539165442, 4.26248153342305032211607885846, 5.77795834298411916829247256156, 6.83508950229635540164151879136, 8.259430965560902117938637901779, 8.724989337689837375123592810194, 9.663395962957927789884371756961, 10.86346312755941098979034917844
