| L(s) = 1 | + 10.2·2-s − 81.6·3-s − 22.5·4-s + 313.·5-s − 838.·6-s + 343·7-s − 1.54e3·8-s + 4.48e3·9-s + 3.21e3·10-s + 4.58e3·11-s + 1.83e3·12-s − 2.19e3·13-s + 3.52e3·14-s − 2.55e4·15-s − 1.29e4·16-s − 1.83e4·17-s + 4.60e4·18-s + 1.35e4·19-s − 7.05e3·20-s − 2.80e4·21-s + 4.71e4·22-s − 5.20e4·23-s + 1.26e5·24-s + 1.98e4·25-s − 2.25e4·26-s − 1.87e5·27-s − 7.72e3·28-s + ⋯ |
| L(s) = 1 | + 0.907·2-s − 1.74·3-s − 0.176·4-s + 1.12·5-s − 1.58·6-s + 0.377·7-s − 1.06·8-s + 2.04·9-s + 1.01·10-s + 1.03·11-s + 0.307·12-s − 0.277·13-s + 0.343·14-s − 1.95·15-s − 0.792·16-s − 0.908·17-s + 1.85·18-s + 0.451·19-s − 0.197·20-s − 0.659·21-s + 0.943·22-s − 0.891·23-s + 1.86·24-s + 0.254·25-s − 0.251·26-s − 1.83·27-s − 0.0665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 - 10.2T + 128T^{2} \) |
| 3 | \( 1 + 81.6T + 2.18e3T^{2} \) |
| 5 | \( 1 - 313.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.58e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.83e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.26e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.35e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.57e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.92e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.70e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.62e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.98e6T + 8.07e13T^{2} \) |
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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
−12.11904122270838157335956369850, −11.42781621551568001451391146952, −10.16094601508599076130742441860, −9.128312580819603783399404458216, −6.83354618707419028568684408695, −5.90055525313729470879474851490, −5.21025079124284140475096666520, −4.08382039946376337335931868279, −1.68019321672298700179106637650, 0,
1.68019321672298700179106637650, 4.08382039946376337335931868279, 5.21025079124284140475096666520, 5.90055525313729470879474851490, 6.83354618707419028568684408695, 9.128312580819603783399404458216, 10.16094601508599076130742441860, 11.42781621551568001451391146952, 12.11904122270838157335956369850
