| L(s) = 1 | + 23.2·2-s + 5.80·3-s + 29.7·4-s + 2.10e3·5-s + 135.·6-s − 2.40e3·7-s − 1.12e4·8-s − 1.96e4·9-s + 4.90e4·10-s + 1.02e3·11-s + 172.·12-s − 2.85e4·13-s − 5.58e4·14-s + 1.22e4·15-s − 2.76e5·16-s − 1.57e5·17-s − 4.57e5·18-s − 5.65e5·19-s + 6.26e4·20-s − 1.39e4·21-s + 2.37e4·22-s − 2.46e5·23-s − 6.51e4·24-s + 2.48e6·25-s − 6.64e5·26-s − 2.28e5·27-s − 7.14e4·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0413·3-s + 0.0580·4-s + 1.50·5-s + 0.0425·6-s − 0.377·7-s − 0.968·8-s − 0.998·9-s + 1.55·10-s + 0.0210·11-s + 0.00240·12-s − 0.277·13-s − 0.388·14-s + 0.0624·15-s − 1.05·16-s − 0.456·17-s − 1.02·18-s − 0.995·19-s + 0.0876·20-s − 0.0156·21-s + 0.0216·22-s − 0.183·23-s − 0.0400·24-s + 1.27·25-s − 0.285·26-s − 0.0826·27-s − 0.0219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 23.2T + 512T^{2} \) |
| 3 | \( 1 - 5.80T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.10e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.02e3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 1.57e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.46e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.71e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.91e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.36e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.84e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.95e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.59e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.39e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.07e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.85e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.35e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.89e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.86e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.83e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.55e8T + 7.60e17T^{2} \) |
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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
−12.10136850109357467547469018135, −10.66882403435088779919325449374, −9.477401019137750247081923874134, −8.638419397052200568961751703147, −6.50793960049654948935504610330, −5.81314034453845501550146186941, −4.76934989065061969976111557136, −3.16719534479068706923807686149, −2.08825319513014781042058671638, 0,
2.08825319513014781042058671638, 3.16719534479068706923807686149, 4.76934989065061969976111557136, 5.81314034453845501550146186941, 6.50793960049654948935504610330, 8.638419397052200568961751703147, 9.477401019137750247081923874134, 10.66882403435088779919325449374, 12.10136850109357467547469018135
