L(s) = 1 | + 6.33·5-s + 19.1·7-s + 41.2·11-s + 39.2·13-s + 59.9·17-s − 41.2·19-s − 53.8·23-s − 84.8·25-s − 65.2·29-s + 64.7·31-s + 121.·35-s + 293.·37-s + 207.·41-s − 377.·43-s + 323.·47-s + 25.4·49-s − 340.·53-s + 261.·55-s + 406.·59-s − 757.·61-s + 248.·65-s + 844.·67-s + 859.·71-s − 789.·73-s + 791.·77-s − 217.·79-s + 101.·83-s + ⋯ |
L(s) = 1 | + 0.567·5-s + 1.03·7-s + 1.13·11-s + 0.837·13-s + 0.855·17-s − 0.498·19-s − 0.487·23-s − 0.678·25-s − 0.417·29-s + 0.375·31-s + 0.587·35-s + 1.30·37-s + 0.789·41-s − 1.34·43-s + 1.00·47-s + 0.0741·49-s − 0.882·53-s + 0.641·55-s + 0.897·59-s − 1.58·61-s + 0.474·65-s + 1.53·67-s + 1.43·71-s − 1.26·73-s + 1.17·77-s − 0.310·79-s + 0.133·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.916627198\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.916627198\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 6.33T + 125T^{2} \) |
| 7 | \( 1 - 19.1T + 343T^{2} \) |
| 11 | \( 1 - 41.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 65.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 207.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 757.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 844.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 859.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 789.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 217.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 101.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
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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
−10.09495890021566546209788882797, −9.314702381683978280278111849802, −8.400881553069678388059525695237, −7.65757063528332426388583359833, −6.39917802288384660339302714165, −5.75026719653185088725618698427, −4.56805206017389704820339630267, −3.61738216645024332781139291175, −2.03615425234351382178290067589, −1.11259601734816875843562038013,
1.11259601734816875843562038013, 2.03615425234351382178290067589, 3.61738216645024332781139291175, 4.56805206017389704820339630267, 5.75026719653185088725618698427, 6.39917802288384660339302714165, 7.65757063528332426388583359833, 8.400881553069678388059525695237, 9.314702381683978280278111849802, 10.09495890021566546209788882797