| L(s) = 1 | − 442.·2-s + 6.56e3·3-s + 6.47e4·4-s − 2.90e6·6-s − 2.47e7·7-s + 2.93e7·8-s + 4.30e7·9-s − 7.66e6·11-s + 4.25e8·12-s + 3.40e9·13-s + 1.09e10·14-s − 2.14e10·16-s + 5.35e10·17-s − 1.90e10·18-s − 1.29e11·19-s − 1.62e11·21-s + 3.39e9·22-s − 2.11e11·23-s + 1.92e11·24-s − 1.50e12·26-s + 2.82e11·27-s − 1.60e12·28-s + 4.13e12·29-s − 5.31e12·31-s + 5.65e12·32-s − 5.03e10·33-s − 2.36e13·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.494·4-s − 0.705·6-s − 1.62·7-s + 0.618·8-s + 0.333·9-s − 0.0107·11-s + 0.285·12-s + 1.15·13-s + 1.98·14-s − 1.24·16-s + 1.86·17-s − 0.407·18-s − 1.75·19-s − 0.938·21-s + 0.0131·22-s − 0.562·23-s + 0.356·24-s − 1.41·26-s + 0.192·27-s − 0.803·28-s + 1.53·29-s − 1.11·31-s + 0.909·32-s − 0.00622·33-s − 2.27·34-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
| \(L(9)\) |
\(\approx\) |
\(0.8737371960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8737371960\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 442.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.47e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.66e6T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.40e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 5.35e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.29e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.11e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 4.13e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.31e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.85e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.57e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.31e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.22e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.74e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 6.31e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 6.08e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.25e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 7.71e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.63e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.45e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.61e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.64e15T + 5.95e33T^{2} \) |
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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
−10.55460419617390333262476931950, −10.01456810457794891162269230165, −8.982339787363553055355070438231, −8.264384253189044889427793672501, −7.04595839684129755508966471530, −5.99505804192014336398860781970, −4.03280324403392691168766691382, −3.04316039387138418198236788905, −1.62927065324616434134238853538, −0.50725288938145783743733092176,
0.50725288938145783743733092176, 1.62927065324616434134238853538, 3.04316039387138418198236788905, 4.03280324403392691168766691382, 5.99505804192014336398860781970, 7.04595839684129755508966471530, 8.264384253189044889427793672501, 8.982339787363553055355070438231, 10.01456810457794891162269230165, 10.55460419617390333262476931950
