| L(s) = 1 | + 3.23·3-s − 3.23·5-s + 7-s + 7.47·9-s − 11-s − 1.23·13-s − 10.4·15-s − 6.47·17-s + 2.76·19-s + 3.23·21-s + 4·23-s + 5.47·25-s + 14.4·27-s + 4.47·29-s + 2·31-s − 3.23·33-s − 3.23·35-s + 10.9·37-s − 4.00·39-s + 6.47·41-s + 1.52·43-s − 24.1·45-s − 2·47-s + 49-s − 20.9·51-s + 0.472·53-s + 3.23·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.44·5-s + 0.377·7-s + 2.49·9-s − 0.301·11-s − 0.342·13-s − 2.70·15-s − 1.56·17-s + 0.634·19-s + 0.706·21-s + 0.834·23-s + 1.09·25-s + 2.78·27-s + 0.830·29-s + 0.359·31-s − 0.563·33-s − 0.546·35-s + 1.79·37-s − 0.640·39-s + 1.01·41-s + 0.232·43-s − 3.60·45-s − 0.291·47-s + 0.142·49-s − 2.93·51-s + 0.0648·53-s + 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189763640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189763640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.320239933204047457567504100982, −7.54263166248512877357244370813, −7.35890106563049715415187539639, −6.40346828831467308434206904196, −4.78234537127146150986268236686, −4.42989790663586766355638151363, −3.66285223200009361512170602448, −2.85731668545147458101611436628, −2.27306179423193761945446246442, −0.906979280347594784601954509677,
0.906979280347594784601954509677, 2.27306179423193761945446246442, 2.85731668545147458101611436628, 3.66285223200009361512170602448, 4.42989790663586766355638151363, 4.78234537127146150986268236686, 6.40346828831467308434206904196, 7.35890106563049715415187539639, 7.54263166248512877357244370813, 8.320239933204047457567504100982
