| L(s) = 1 | − 659.·2-s − 6.56e3·3-s + 3.03e5·4-s + 4.32e6·6-s − 1.59e6·7-s − 1.13e8·8-s + 4.30e7·9-s − 4.47e8·11-s − 1.99e9·12-s + 2.48e9·13-s + 1.05e9·14-s + 3.50e10·16-s + 2.48e10·17-s − 2.83e10·18-s + 8.23e10·19-s + 1.04e10·21-s + 2.94e11·22-s − 6.43e11·23-s + 7.44e11·24-s − 1.63e12·26-s − 2.82e11·27-s − 4.84e11·28-s − 9.82e11·29-s + 3.28e12·31-s − 8.23e12·32-s + 2.93e12·33-s − 1.63e13·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.31·4-s + 1.05·6-s − 0.104·7-s − 2.39·8-s + 0.333·9-s − 0.629·11-s − 1.33·12-s + 0.844·13-s + 0.190·14-s + 2.04·16-s + 0.864·17-s − 0.606·18-s + 1.11·19-s + 0.0605·21-s + 1.14·22-s − 1.71·23-s + 1.38·24-s − 1.53·26-s − 0.192·27-s − 0.242·28-s − 0.364·29-s + 0.692·31-s − 1.32·32-s + 0.363·33-s − 1.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 659.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 1.59e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.47e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.48e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.48e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.23e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 6.43e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 9.82e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.28e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.63e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.33e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 9.83e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.62e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.40e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 9.80e13T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.37e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.85e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 6.17e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.30e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.27e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.42e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.77e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.09e17T + 5.95e33T^{2} \) |
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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
−10.32989774505370643497130988497, −9.769840985603393586469956427618, −8.441817305713510672001413719969, −7.65154711424314819317798832238, −6.52754472913296319570206905257, −5.45394334447855639385894695866, −3.41030701197852577735793242273, −1.94917488041806777527201370292, −0.967203486340792901661110955163, 0,
0.967203486340792901661110955163, 1.94917488041806777527201370292, 3.41030701197852577735793242273, 5.45394334447855639385894695866, 6.52754472913296319570206905257, 7.65154711424314819317798832238, 8.441817305713510672001413719969, 9.769840985603393586469956427618, 10.32989774505370643497130988497
