| L(s) = 1 | + 564.·2-s − 1.71e4·3-s + 1.87e5·4-s − 3.90e5·5-s − 9.70e6·6-s − 2.21e7·7-s + 3.21e7·8-s + 1.66e8·9-s − 2.20e8·10-s − 7.27e8·11-s − 3.22e9·12-s + 3.52e9·13-s − 1.24e10·14-s + 6.71e9·15-s − 6.48e9·16-s + 3.37e10·17-s + 9.37e10·18-s − 7.67e10·19-s − 7.34e10·20-s + 3.80e11·21-s − 4.11e11·22-s + 5.48e10·23-s − 5.51e11·24-s + 1.52e11·25-s + 1.99e12·26-s − 6.33e11·27-s − 4.15e12·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 1.51·3-s + 1.43·4-s − 0.447·5-s − 2.35·6-s − 1.45·7-s + 0.676·8-s + 1.28·9-s − 0.697·10-s − 1.02·11-s − 2.16·12-s + 1.19·13-s − 2.26·14-s + 0.676·15-s − 0.377·16-s + 1.17·17-s + 2.00·18-s − 1.03·19-s − 0.641·20-s + 2.19·21-s − 1.59·22-s + 0.145·23-s − 1.02·24-s + 0.200·25-s + 1.87·26-s − 0.431·27-s − 2.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 - 564.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.71e4T + 1.29e8T^{2} \) |
| 7 | \( 1 + 2.21e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.27e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.52e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.37e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 7.67e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.48e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.43e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 8.21e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.72e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 7.17e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 2.55e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 3.00e12T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.88e13T + 2.05e29T^{2} \) |
| 59 | \( 1 + 8.49e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.42e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.19e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.19e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 9.91e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 7.23e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.82e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.95e15T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.10e17T + 5.95e33T^{2} \) |
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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
−18.70666582729506741072839698224, −16.50233143307076047376348149703, −15.56090819588696548742524599750, −13.18359663416324163419100046435, −12.23855589504758102371847651696, −10.71618461255052101354006384601, −6.53294734135033074838806172706, −5.41475170613142287077539820900, −3.54383608217808453420825606638, 0,
3.54383608217808453420825606638, 5.41475170613142287077539820900, 6.53294734135033074838806172706, 10.71618461255052101354006384601, 12.23855589504758102371847651696, 13.18359663416324163419100046435, 15.56090819588696548742524599750, 16.50233143307076047376348149703, 18.70666582729506741072839698224
