| L(s) = 1 | − 2·2-s + 4·4-s + 8.06·5-s + 26.0·7-s − 8·8-s − 16.1·10-s + 3.26·13-s − 52.1·14-s + 16·16-s − 20.8·17-s + 125.·19-s + 32.2·20-s − 97.8·23-s − 60.0·25-s − 6.52·26-s + 104.·28-s − 263.·29-s + 199.·31-s − 32·32-s + 41.7·34-s + 210.·35-s + 365.·37-s − 251.·38-s − 64.4·40-s + 273.·41-s − 388.·43-s + 195.·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.721·5-s + 1.40·7-s − 0.353·8-s − 0.509·10-s + 0.0696·13-s − 0.995·14-s + 0.250·16-s − 0.297·17-s + 1.52·19-s + 0.360·20-s − 0.886·23-s − 0.480·25-s − 0.0492·26-s + 0.704·28-s − 1.68·29-s + 1.15·31-s − 0.176·32-s + 0.210·34-s + 1.01·35-s + 1.62·37-s − 1.07·38-s − 0.254·40-s + 1.04·41-s − 1.37·43-s + 0.627·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.393891863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393891863\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 8.06T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 13 | \( 1 - 3.26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 51.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 412.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 164.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 516.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 241.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 72.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−8.724903687948606644912505989820, −7.82519822573404537256397196601, −7.56252575450404111362186573255, −6.34057663536787906604920972377, −5.62793746615753788021806831082, −4.86317670349778468747921821427, −3.77278142683760007220205769504, −2.43977293484378959767511720666, −1.73641210411857431143493427359, −0.814393950851308514054367151952,
0.814393950851308514054367151952, 1.73641210411857431143493427359, 2.43977293484378959767511720666, 3.77278142683760007220205769504, 4.86317670349778468747921821427, 5.62793746615753788021806831082, 6.34057663536787906604920972377, 7.56252575450404111362186573255, 7.82519822573404537256397196601, 8.724903687948606644912505989820
