| L(s) = 1 | + 19.3·2-s + 27·3-s + 247.·4-s + 336.·5-s + 523.·6-s + 879.·7-s + 2.31e3·8-s + 729·9-s + 6.51e3·10-s + 6.68e3·12-s − 460.·13-s + 1.70e4·14-s + 9.07e3·15-s + 1.32e4·16-s − 2.49e4·17-s + 1.41e4·18-s + 2.47e3·19-s + 8.31e4·20-s + 2.37e4·21-s + 1.05e5·23-s + 6.25e4·24-s + 3.47e4·25-s − 8.92e3·26-s + 1.96e4·27-s + 2.17e5·28-s + 1.28e5·29-s + 1.75e5·30-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s + 1.20·5-s + 0.988·6-s + 0.968·7-s + 1.59·8-s + 0.333·9-s + 2.05·10-s + 1.11·12-s − 0.0581·13-s + 1.65·14-s + 0.694·15-s + 0.806·16-s − 1.23·17-s + 0.570·18-s + 0.0826·19-s + 2.32·20-s + 0.559·21-s + 1.80·23-s + 0.923·24-s + 0.445·25-s − 0.0996·26-s + 0.192·27-s + 1.87·28-s + 0.975·29-s + 1.18·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(11.80358941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.80358941\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 19.3T + 128T^{2} \) |
| 5 | \( 1 - 336.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 879.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 460.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.47e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.05e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.94e3T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.33e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.71e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.48e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.71e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.67e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.79e4T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.03e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.10e6T + 8.07e13T^{2} \) |
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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
−10.53108800293718290711319427893, −9.322269686744258271826244417379, −8.315825664525592497442183436483, −6.94819930283901460451677213228, −6.25685315901166306378928972803, −5.01576969193143364892280461732, −4.63594095738563932512303784474, −3.16983346796733487431309523693, −2.31212456431414682320771515910, −1.44217453290855374541879106503,
1.44217453290855374541879106503, 2.31212456431414682320771515910, 3.16983346796733487431309523693, 4.63594095738563932512303784474, 5.01576969193143364892280461732, 6.25685315901166306378928972803, 6.94819930283901460451677213228, 8.315825664525592497442183436483, 9.322269686744258271826244417379, 10.53108800293718290711319427893
