| L(s) = 1 | − 25.6·3-s − 28.7·5-s + 414.·9-s + 270.·11-s − 300.·13-s + 737.·15-s − 613.·17-s − 1.70e3·19-s − 3.18e3·23-s − 2.29e3·25-s − 4.40e3·27-s + 4.29e3·29-s + 2.02e3·31-s − 6.92e3·33-s + 5.15e3·37-s + 7.71e3·39-s + 7.14e3·41-s + 1.95e4·43-s − 1.19e4·45-s + 1.99e4·47-s + 1.57e4·51-s + 3.94e3·53-s − 7.76e3·55-s + 4.36e4·57-s − 2.97e4·59-s + 5.05e4·61-s + 8.64e3·65-s + ⋯ |
| L(s) = 1 | − 1.64·3-s − 0.514·5-s + 1.70·9-s + 0.673·11-s − 0.493·13-s + 0.846·15-s − 0.514·17-s − 1.08·19-s − 1.25·23-s − 0.735·25-s − 1.16·27-s + 0.949·29-s + 0.379·31-s − 1.10·33-s + 0.618·37-s + 0.811·39-s + 0.663·41-s + 1.61·43-s − 0.878·45-s + 1.32·47-s + 0.846·51-s + 0.193·53-s − 0.346·55-s + 1.77·57-s − 1.11·59-s + 1.73·61-s + 0.253·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 25.6T + 243T^{2} \) |
| 5 | \( 1 + 28.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 613.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.94e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.36e4T + 8.58e9T^{2} \) |
| 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
−9.268534523482326171955663128888, −8.121340049307635827705013313140, −7.16710195815937774521569249382, −6.31241041136095772976598767488, −5.75690042585202149226125922469, −4.49346847522710412689369171761, −4.09069814619518151244127467890, −2.27166446622554228351210149928, −0.886221142913795246061684425110, 0,
0.886221142913795246061684425110, 2.27166446622554228351210149928, 4.09069814619518151244127467890, 4.49346847522710412689369171761, 5.75690042585202149226125922469, 6.31241041136095772976598767488, 7.16710195815937774521569249382, 8.121340049307635827705013313140, 9.268534523482326171955663128888
