| L(s) = 1 | − 52.4·3-s − 199.·5-s + 343·7-s + 559.·9-s − 1.21e3·11-s − 6.44e3·13-s + 1.04e4·15-s + 2.67e4·17-s − 1.49e4·19-s − 1.79e4·21-s − 7.35e4·23-s − 3.84e4·25-s + 8.52e4·27-s + 1.06e5·29-s + 6.68e4·31-s + 6.38e4·33-s − 6.83e4·35-s + 4.79e5·37-s + 3.37e5·39-s − 6.44e5·41-s − 1.45e5·43-s − 1.11e5·45-s + 1.01e6·47-s + 1.17e5·49-s − 1.40e6·51-s − 3.42e4·53-s + 2.42e5·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s − 0.712·5-s + 0.377·7-s + 0.255·9-s − 0.276·11-s − 0.814·13-s + 0.798·15-s + 1.32·17-s − 0.498·19-s − 0.423·21-s − 1.25·23-s − 0.492·25-s + 0.834·27-s + 0.810·29-s + 0.402·31-s + 0.309·33-s − 0.269·35-s + 1.55·37-s + 0.912·39-s − 1.46·41-s − 0.278·43-s − 0.182·45-s + 1.42·47-s + 0.142·49-s − 1.48·51-s − 0.0315·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 + 52.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 199.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.42e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.72e7T + 8.07e13T^{2} \) |
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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
−9.792344650203084942746359083932, −8.289444645509624351905260963871, −7.72948972812547488536237981551, −6.58881186931987587118151143591, −5.63791751515881964954358531708, −4.84613614095476282097981693385, −3.82636698045535018116626317221, −2.42365064276245658957870385685, −0.911263707181631721031826494789, 0,
0.911263707181631721031826494789, 2.42365064276245658957870385685, 3.82636698045535018116626317221, 4.84613614095476282097981693385, 5.63791751515881964954358531708, 6.58881186931987587118151143591, 7.72948972812547488536237981551, 8.289444645509624351905260963871, 9.792344650203084942746359083932
