L(s) = 1 | + 16.1·2-s + 27·3-s + 131.·4-s − 519.·5-s + 435.·6-s − 439.·7-s + 64.2·8-s + 729·9-s − 8.37e3·10-s − 1.23e3·11-s + 3.56e3·12-s − 886.·13-s − 7.08e3·14-s − 1.40e4·15-s − 1.58e4·16-s − 3.32e4·17-s + 1.17e4·18-s + 2.42e4·19-s − 6.85e4·20-s − 1.18e4·21-s − 1.99e4·22-s − 1.21e4·23-s + 1.73e3·24-s + 1.91e5·25-s − 1.42e4·26-s + 1.96e4·27-s − 5.79e4·28-s + ⋯ |
L(s) = 1 | + 1.42·2-s + 0.577·3-s + 1.03·4-s − 1.85·5-s + 0.822·6-s − 0.484·7-s + 0.0443·8-s + 0.333·9-s − 2.64·10-s − 0.280·11-s + 0.595·12-s − 0.111·13-s − 0.689·14-s − 1.07·15-s − 0.967·16-s − 1.64·17-s + 0.475·18-s + 0.810·19-s − 1.91·20-s − 0.279·21-s − 0.400·22-s − 0.208·23-s + 0.0256·24-s + 2.45·25-s − 0.159·26-s + 0.192·27-s − 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
good | 2 | \( 1 - 16.1T + 128T^{2} \) |
| 5 | \( 1 + 519.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 439.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 886.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.42e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.63e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.83e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 7.26e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.02e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.15e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.15e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.13e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 6.99e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.69e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.31e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.25e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.21e7T + 8.07e13T^{2} \) |
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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
−12.81954688544024344554895970520, −11.94532843950750774336864336087, −11.01122518147435305558549289145, −9.061813096883941239502051841991, −7.77444249286341595585804715998, −6.63814897092251041217507209205, −4.75029310677354342908425425876, −3.84709182614320379084560114337, −2.84054475074081385744146118968, 0,
2.84054475074081385744146118968, 3.84709182614320379084560114337, 4.75029310677354342908425425876, 6.63814897092251041217507209205, 7.77444249286341595585804715998, 9.061813096883941239502051841991, 11.01122518147435305558549289145, 11.94532843950750774336864336087, 12.81954688544024344554895970520