| L(s) = 1 | − 4·2-s + 26.5·3-s + 16·4-s + 25·5-s − 106.·6-s + 257.·7-s − 64·8-s + 462.·9-s − 100·10-s + 382.·11-s + 425.·12-s + 152.·13-s − 1.03e3·14-s + 664.·15-s + 256·16-s + 437.·17-s − 1.85e3·18-s − 2.05e3·19-s + 400·20-s + 6.85e3·21-s − 1.52e3·22-s − 529·23-s − 1.70e3·24-s + 625·25-s − 611.·26-s + 5.84e3·27-s + 4.12e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.70·3-s + 0.5·4-s + 0.447·5-s − 1.20·6-s + 1.98·7-s − 0.353·8-s + 1.90·9-s − 0.316·10-s + 0.952·11-s + 0.852·12-s + 0.251·13-s − 1.40·14-s + 0.762·15-s + 0.250·16-s + 0.367·17-s − 1.34·18-s − 1.30·19-s + 0.223·20-s + 3.39·21-s − 0.673·22-s − 0.208·23-s − 0.602·24-s + 0.200·25-s − 0.177·26-s + 1.54·27-s + 0.994·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.886994438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.886994438\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 - 26.5T + 243T^{2} \) |
| 7 | \( 1 - 257.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 382.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 437.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.05e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.13e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−11.11151578479588898268534155097, −10.15241664971668333643978968920, −9.033925305323271989508075686435, −8.458156446195016030767542504395, −7.85090226506196515082101010132, −6.66510802785288873600071190689, −4.85433946833117576646713026073, −3.56674647041139659420305073685, −1.96990544035844286340248081941, −1.54850017254776971762569449061,
1.54850017254776971762569449061, 1.96990544035844286340248081941, 3.56674647041139659420305073685, 4.85433946833117576646713026073, 6.66510802785288873600071190689, 7.85090226506196515082101010132, 8.458156446195016030767542504395, 9.033925305323271989508075686435, 10.15241664971668333643978968920, 11.11151578479588898268534155097
