| L(s) = 1 | − 2.93·2-s + 0.640·4-s − 2.56·5-s − 7·7-s + 21.6·8-s + 7.55·10-s − 0.514·11-s + 65.4·13-s + 20.5·14-s − 68.7·16-s − 3.37·17-s − 123.·19-s − 1.64·20-s + 1.51·22-s − 25.1·23-s − 118.·25-s − 192.·26-s − 4.48·28-s + 237.·29-s − 89.9·31-s + 28.9·32-s + 9.92·34-s + 17.9·35-s − 67.4·37-s + 364.·38-s − 55.5·40-s + 287.·41-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.0800·4-s − 0.229·5-s − 0.377·7-s + 0.956·8-s + 0.238·10-s − 0.0140·11-s + 1.39·13-s + 0.392·14-s − 1.07·16-s − 0.0481·17-s − 1.49·19-s − 0.0183·20-s + 0.0146·22-s − 0.227·23-s − 0.947·25-s − 1.45·26-s − 0.0302·28-s + 1.52·29-s − 0.520·31-s + 0.159·32-s + 0.0500·34-s + 0.0868·35-s − 0.299·37-s + 1.55·38-s − 0.219·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7680519938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7680519938\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.93T + 8T^{2} \) |
| 5 | \( 1 + 2.56T + 125T^{2} \) |
| 11 | \( 1 + 0.514T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.37T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 89.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 435.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 516.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 251.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 461.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 532.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 360.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 762.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 944.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−10.33016686355324685851421520244, −9.358307267021173678931155478627, −8.541748055956210050494889639128, −8.055431540826428377014322528464, −6.85932355854094028469383136568, −6.00980882650627626644207627525, −4.55740884450196740327020717962, −3.62341818447250313227293785005, −1.95900319766847864869681448614, −0.62278174341597328926851622858,
0.62278174341597328926851622858, 1.95900319766847864869681448614, 3.62341818447250313227293785005, 4.55740884450196740327020717962, 6.00980882650627626644207627525, 6.85932355854094028469383136568, 8.055431540826428377014322528464, 8.541748055956210050494889639128, 9.358307267021173678931155478627, 10.33016686355324685851421520244
