| L(s) = 1 | + 2.35·2-s + 1.91·3-s + 3.53·4-s + 4.51·6-s − 0.845·7-s + 3.62·8-s + 0.675·9-s − 1.11·11-s + 6.78·12-s + 2.84·13-s − 1.99·14-s + 1.44·16-s − 3.11·17-s + 1.59·18-s + 4.69·19-s − 1.62·21-s − 2.62·22-s + 4.28·23-s + 6.94·24-s + 6.69·26-s − 4.45·27-s − 2.99·28-s − 2.93·29-s + 31-s − 3.84·32-s − 2.13·33-s − 7.33·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.10·3-s + 1.76·4-s + 1.84·6-s − 0.319·7-s + 1.28·8-s + 0.225·9-s − 0.335·11-s + 1.95·12-s + 0.789·13-s − 0.532·14-s + 0.361·16-s − 0.755·17-s + 0.374·18-s + 1.07·19-s − 0.353·21-s − 0.558·22-s + 0.892·23-s + 1.41·24-s + 1.31·26-s − 0.857·27-s − 0.565·28-s − 0.544·29-s + 0.179·31-s − 0.678·32-s − 0.371·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.046046213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.046046213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.91T + 3T^{2} \) |
| 7 | \( 1 + 0.845T + 7T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 0.658T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 + 0.847T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 - 6.95T + 73T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 - 7.38T + 83T^{2} \) |
| 89 | \( 1 - 9.22T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−10.54721478947834818558889350311, −9.292528567961548609394949751840, −8.626460418584217426521071233837, −7.50370818595951847163522582466, −6.67723909546139142738386033590, −5.67312533097120110528077545658, −4.80880842260891333398642698619, −3.58038462814381200282414729121, −3.16491095427244489681280194723, −2.02682888086460042573668803552,
2.02682888086460042573668803552, 3.16491095427244489681280194723, 3.58038462814381200282414729121, 4.80880842260891333398642698619, 5.67312533097120110528077545658, 6.67723909546139142738386033590, 7.50370818595951847163522582466, 8.626460418584217426521071233837, 9.292528567961548609394949751840, 10.54721478947834818558889350311
