| L(s) = 1 | − 8·2-s + 12.8·3-s + 64·4-s + 260.·5-s − 102.·6-s − 982.·7-s − 512·8-s − 2.02e3·9-s − 2.08e3·10-s − 4.72e3·11-s + 822.·12-s + 7.85e3·14-s + 3.34e3·15-s + 4.09e3·16-s + 3.41e4·17-s + 1.61e4·18-s + 4.46e4·19-s + 1.66e4·20-s − 1.26e4·21-s + 3.78e4·22-s + 1.83e4·23-s − 6.58e3·24-s − 1.05e4·25-s − 5.41e4·27-s − 6.28e4·28-s + 8.78e4·29-s − 2.67e4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.274·3-s + 0.5·4-s + 0.930·5-s − 0.194·6-s − 1.08·7-s − 0.353·8-s − 0.924·9-s − 0.657·10-s − 1.07·11-s + 0.137·12-s + 0.765·14-s + 0.255·15-s + 0.250·16-s + 1.68·17-s + 0.653·18-s + 1.49·19-s + 0.465·20-s − 0.297·21-s + 0.756·22-s + 0.313·23-s − 0.0971·24-s − 0.134·25-s − 0.528·27-s − 0.541·28-s + 0.668·29-s − 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 12.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 260.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 982.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.72e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.41e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.46e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.78e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.83e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.80e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.57e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.45e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.05e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02637436676247300319759490105, −9.065235767365672328815322331036, −8.102893045820010849678597795686, −7.15300758688343569905605890332, −5.92066167551446474240074015154, −5.37989182882182797336469423447, −3.18533183675029684076361656517, −2.73444955533363656441272339489, −1.23749999495705659115873869795, 0,
1.23749999495705659115873869795, 2.73444955533363656441272339489, 3.18533183675029684076361656517, 5.37989182882182797336469423447, 5.92066167551446474240074015154, 7.15300758688343569905605890332, 8.102893045820010849678597795686, 9.065235767365672328815322331036, 10.02637436676247300319759490105
