L(s) = 1 | − 3.80·2-s − 76.5·3-s − 113.·4-s − 339.·5-s + 290.·6-s + 343·7-s + 917.·8-s + 3.67e3·9-s + 1.29e3·10-s + 3.39e3·11-s + 8.69e3·12-s − 2.19e3·13-s − 1.30e3·14-s + 2.59e4·15-s + 1.10e4·16-s + 8.09e3·17-s − 1.39e4·18-s − 4.96e4·19-s + 3.85e4·20-s − 2.62e4·21-s − 1.29e4·22-s + 2.06e4·23-s − 7.02e4·24-s + 3.71e4·25-s + 8.34e3·26-s − 1.13e5·27-s − 3.89e4·28-s + ⋯ |
L(s) = 1 | − 0.335·2-s − 1.63·3-s − 0.887·4-s − 1.21·5-s + 0.549·6-s + 0.377·7-s + 0.633·8-s + 1.67·9-s + 0.407·10-s + 0.770·11-s + 1.45·12-s − 0.277·13-s − 0.126·14-s + 1.98·15-s + 0.674·16-s + 0.399·17-s − 0.563·18-s − 1.66·19-s + 1.07·20-s − 0.618·21-s − 0.258·22-s + 0.353·23-s − 1.03·24-s + 0.475·25-s + 0.0931·26-s − 1.11·27-s − 0.335·28-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 | \( 1 + 3.80T + 128T^{2} \) |
| 3 | \( 1 + 76.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 339.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.39e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 8.09e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.96e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.25e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.04e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.24e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.73e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.82e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.70e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.26e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.03e7T + 8.07e13T^{2} \) |
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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
−12.03717797773810890765819843767, −11.12201132046029967455700527574, −10.22658292179547027813625033623, −8.742902181345339337182100826966, −7.58312240079118369304208126311, −6.26296463379932050325315901245, −4.80037475847453748240447208417, −4.10703250810770656976087154394, −0.995886737109739991821878512394, 0,
0.995886737109739991821878512394, 4.10703250810770656976087154394, 4.80037475847453748240447208417, 6.26296463379932050325315901245, 7.58312240079118369304208126311, 8.742902181345339337182100826966, 10.22658292179547027813625033623, 11.12201132046029967455700527574, 12.03717797773810890765819843767