| L(s) = 1 | + 21.8·3-s + 25·5-s − 49·7-s + 235.·9-s − 502.·11-s − 858.·13-s + 547.·15-s − 1.49e3·17-s + 257.·19-s − 1.07e3·21-s − 4.00e3·23-s + 625·25-s − 158.·27-s − 4.30e3·29-s + 7.53e3·31-s − 1.10e4·33-s − 1.22e3·35-s + 9.02e3·37-s − 1.87e4·39-s + 659.·41-s − 1.13e4·43-s + 5.89e3·45-s − 1.44e4·47-s + 2.40e3·49-s − 3.27e4·51-s + 2.58e4·53-s − 1.25e4·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 0.447·5-s − 0.377·7-s + 0.970·9-s − 1.25·11-s − 1.40·13-s + 0.627·15-s − 1.25·17-s + 0.163·19-s − 0.530·21-s − 1.57·23-s + 0.200·25-s − 0.0418·27-s − 0.949·29-s + 1.40·31-s − 1.75·33-s − 0.169·35-s + 1.08·37-s − 1.97·39-s + 0.0613·41-s − 0.935·43-s + 0.433·45-s − 0.954·47-s + 0.142·49-s − 1.76·51-s + 1.26·53-s − 0.560·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 21.8T + 243T^{2} \) |
| 11 | \( 1 + 502.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 858.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 257.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.00e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 659.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.30e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.53e5T + 8.58e9T^{2} \) |
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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
−9.989926544160370132370361190853, −9.773044483547376559140304989930, −8.525062050498309552010755000038, −7.83880408036866029092187279991, −6.78911766124088064142510895916, −5.37378632764922417859751594880, −4.10714946242204420527586907919, −2.66489186607734107654184784898, −2.20748459355736745805899095861, 0,
2.20748459355736745805899095861, 2.66489186607734107654184784898, 4.10714946242204420527586907919, 5.37378632764922417859751594880, 6.78911766124088064142510895916, 7.83880408036866029092187279991, 8.525062050498309552010755000038, 9.773044483547376559140304989930, 9.989926544160370132370361190853
