| L(s) = 1 | − 10.7·3-s − 106.·5-s − 126.·9-s + 93.4·11-s − 661.·13-s + 1.14e3·15-s − 455.·17-s + 1.10e3·19-s − 748.·23-s + 8.13e3·25-s + 3.98e3·27-s + 2.80e3·29-s − 359.·31-s − 1.00e3·33-s − 6.81e3·37-s + 7.12e3·39-s − 2.31e3·41-s + 1.99e4·43-s + 1.34e4·45-s − 1.42e4·47-s + 4.91e3·51-s + 2.61e4·53-s − 9.91e3·55-s − 1.19e4·57-s + 4.90e3·59-s + 1.32e4·61-s + 7.01e4·65-s + ⋯ |
| L(s) = 1 | − 0.691·3-s − 1.89·5-s − 0.522·9-s + 0.232·11-s − 1.08·13-s + 1.31·15-s − 0.382·17-s + 0.703·19-s − 0.294·23-s + 2.60·25-s + 1.05·27-s + 0.619·29-s − 0.0671·31-s − 0.160·33-s − 0.818·37-s + 0.750·39-s − 0.215·41-s + 1.64·43-s + 0.990·45-s − 0.938·47-s + 0.264·51-s + 1.27·53-s − 0.441·55-s − 0.486·57-s + 0.183·59-s + 0.454·61-s + 2.05·65-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 | \( 1 + 10.7T + 243T^{2} \) |
| 5 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 93.4T + 1.61e5T^{2} \) |
| 13 | \( 1 + 661.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 455.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 748.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 359.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.42e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.90e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.23e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.33e4T + 8.58e9T^{2} \) |
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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
−8.938540847467628183797002325972, −8.175904435272867054406961562599, −7.36140185955878622454738884860, −6.68243769477460703064187798288, −5.38868077315536056409838861131, −4.60682136051758522715473113066, −3.69364195861943475265032937162, −2.66194614803501803473022161238, −0.792786070215387526813688467770, 0,
0.792786070215387526813688467770, 2.66194614803501803473022161238, 3.69364195861943475265032937162, 4.60682136051758522715473113066, 5.38868077315536056409838861131, 6.68243769477460703064187798288, 7.36140185955878622454738884860, 8.175904435272867054406961562599, 8.938540847467628183797002325972
