| L(s) = 1 | − 2.18·2-s + 2.66·3-s + 2.78·4-s − 5.83·6-s − 7-s − 1.70·8-s + 4.11·9-s − 3.38·11-s + 7.41·12-s − 0.0686·13-s + 2.18·14-s − 1.82·16-s − 2.64·17-s − 8.99·18-s + 3.27·19-s − 2.66·21-s + 7.40·22-s + 23-s − 4.55·24-s + 0.150·26-s + 2.96·27-s − 2.78·28-s − 0.498·29-s − 0.846·31-s + 7.41·32-s − 9.03·33-s + 5.79·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.53·3-s + 1.39·4-s − 2.38·6-s − 0.377·7-s − 0.604·8-s + 1.37·9-s − 1.02·11-s + 2.14·12-s − 0.0190·13-s + 0.584·14-s − 0.456·16-s − 0.642·17-s − 2.11·18-s + 0.751·19-s − 0.581·21-s + 1.57·22-s + 0.208·23-s − 0.930·24-s + 0.0294·26-s + 0.570·27-s − 0.525·28-s − 0.0926·29-s − 0.152·31-s + 1.30·32-s − 1.57·33-s + 0.993·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306920283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306920283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 2.66T + 3T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 0.0686T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 29 | \( 1 + 0.498T + 29T^{2} \) |
| 31 | \( 1 + 0.846T + 31T^{2} \) |
| 37 | \( 1 - 2.34T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 0.0435T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 8.53T + 61T^{2} \) |
| 67 | \( 1 - 8.46T + 67T^{2} \) |
| 71 | \( 1 + 0.438T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 + 9.44T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 - 6.71T + 97T^{2} \) |
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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
−8.613508935815569861640742838864, −7.85638697620527095125001350351, −7.43452966963753529634743763415, −6.78268178159806037442386866564, −5.61991473192644510812475982576, −4.47552479958727571463656568767, −3.45981289200928368402441051749, −2.57754111005712293689574996417, −2.06655016037952633097703197159, −0.75027390374802439012494657002,
0.75027390374802439012494657002, 2.06655016037952633097703197159, 2.57754111005712293689574996417, 3.45981289200928368402441051749, 4.47552479958727571463656568767, 5.61991473192644510812475982576, 6.78268178159806037442386866564, 7.43452966963753529634743763415, 7.85638697620527095125001350351, 8.613508935815569861640742838864
