| L(s) = 1 | − 0.796·2-s + 10.5·3-s − 31.3·4-s − 25·5-s − 8.42·6-s + 50.4·8-s − 131.·9-s + 19.9·10-s + 525.·11-s − 331.·12-s + 819.·13-s − 264.·15-s + 963.·16-s − 921.·17-s + 104.·18-s − 696.·19-s + 784.·20-s − 418.·22-s + 2.38e3·23-s + 533.·24-s + 625·25-s − 652.·26-s − 3.95e3·27-s − 6.20e3·29-s + 210.·30-s − 8.01e3·31-s − 2.38e3·32-s + ⋯ |
| L(s) = 1 | − 0.140·2-s + 0.678·3-s − 0.980·4-s − 0.447·5-s − 0.0955·6-s + 0.278·8-s − 0.539·9-s + 0.0629·10-s + 1.30·11-s − 0.664·12-s + 1.34·13-s − 0.303·15-s + 0.940·16-s − 0.773·17-s + 0.0760·18-s − 0.442·19-s + 0.438·20-s − 0.184·22-s + 0.938·23-s + 0.189·24-s + 0.200·25-s − 0.189·26-s − 1.04·27-s − 1.36·29-s + 0.0427·30-s − 1.49·31-s − 0.411·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.796T + 32T^{2} \) |
| 3 | \( 1 - 10.5T + 243T^{2} \) |
| 11 | \( 1 - 525.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 819.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 921.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 696.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.32e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.26e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.35e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.14e4T + 8.58e9T^{2} \) |
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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
−10.83737086734376389008228315712, −9.274418277148631864371610138734, −8.911377162647097495760981801099, −8.154251433732314968953315595313, −6.83013128016108164559027940572, −5.52312059150619166021419666986, −4.06318157292085137732634478981, −3.44864088181363275860441974354, −1.53698927422100383023646048256, 0,
1.53698927422100383023646048256, 3.44864088181363275860441974354, 4.06318157292085137732634478981, 5.52312059150619166021419666986, 6.83013128016108164559027940572, 8.154251433732314968953315595313, 8.911377162647097495760981801099, 9.274418277148631864371610138734, 10.83737086734376389008228315712
