| L(s) = 1 | − 20.3·2-s + 286.·4-s − 125·5-s − 343·7-s − 3.23e3·8-s + 2.54e3·10-s − 3.07e3·11-s + 208.·13-s + 6.98e3·14-s + 2.90e4·16-s − 9.53e3·17-s + 2.87e4·19-s − 3.58e4·20-s + 6.25e4·22-s + 5.94e4·23-s + 1.56e4·25-s − 4.25e3·26-s − 9.83e4·28-s − 8.18e3·29-s − 1.30e5·31-s − 1.78e5·32-s + 1.94e5·34-s + 4.28e4·35-s − 2.79e5·37-s − 5.84e5·38-s + 4.03e5·40-s + 3.49e5·41-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.23·4-s − 0.447·5-s − 0.377·7-s − 2.23·8-s + 0.804·10-s − 0.695·11-s + 0.0263·13-s + 0.680·14-s + 1.77·16-s − 0.470·17-s + 0.960·19-s − 1.00·20-s + 1.25·22-s + 1.01·23-s + 0.199·25-s − 0.0474·26-s − 0.846·28-s − 0.0623·29-s − 0.788·31-s − 0.964·32-s + 0.847·34-s + 0.169·35-s − 0.908·37-s − 1.72·38-s + 0.997·40-s + 0.791·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 20.3T + 128T^{2} \) |
| 11 | \( 1 + 3.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 208.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 9.53e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.87e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.18e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.30e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.49e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.72e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.10e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.06e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.77e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.43e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.06e6T + 8.07e13T^{2} \) |
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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
−9.806456686826811870558725589896, −9.077867571853383537477842744356, −8.185369090143779743039298296650, −7.37161657174663007921444155895, −6.61975226839881070077700598918, −5.21405397770189648553661250253, −3.39757500975577929141339328057, −2.27726281073138559322683620191, −0.962903306205940770838282234545, 0,
0.962903306205940770838282234545, 2.27726281073138559322683620191, 3.39757500975577929141339328057, 5.21405397770189648553661250253, 6.61975226839881070077700598918, 7.37161657174663007921444155895, 8.185369090143779743039298296650, 9.077867571853383537477842744356, 9.806456686826811870558725589896
