| L(s) = 1 | + 3·3-s + 5.56·5-s + 9·9-s + 13.9·11-s + 38.6·13-s + 16.6·15-s + 43.4·17-s + 109.·19-s + 74.8·23-s − 94.0·25-s + 27·27-s − 72.3·29-s − 64.0·31-s + 41.7·33-s + 188.·37-s + 116.·39-s − 24.7·41-s + 243.·43-s + 50.0·45-s + 620.·47-s + 130.·51-s − 287.·53-s + 77.4·55-s + 327.·57-s − 525.·59-s − 383.·61-s + 215.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.497·5-s + 0.333·9-s + 0.381·11-s + 0.825·13-s + 0.287·15-s + 0.620·17-s + 1.31·19-s + 0.678·23-s − 0.752·25-s + 0.192·27-s − 0.463·29-s − 0.371·31-s + 0.220·33-s + 0.838·37-s + 0.476·39-s − 0.0944·41-s + 0.864·43-s + 0.165·45-s + 1.92·47-s + 0.358·51-s − 0.745·53-s + 0.189·55-s + 0.760·57-s − 1.15·59-s − 0.804·61-s + 0.410·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.880414961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.880414961\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5.56T + 125T^{2} \) |
| 11 | \( 1 - 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 383.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 79.2T + 9.12e5T^{2} \) |
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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
−8.877311926556311718157585216342, −7.69303512813081196216099799400, −7.37616857535695739731354432730, −6.14231053055773284935287291528, −5.67258523172997287200256595533, −4.56442995314931697813478200805, −3.61825740202547922492559899269, −2.90005117890120903478917245731, −1.74249318066661409041432220034, −0.912029186720007407272693273454,
0.912029186720007407272693273454, 1.74249318066661409041432220034, 2.90005117890120903478917245731, 3.61825740202547922492559899269, 4.56442995314931697813478200805, 5.67258523172997287200256595533, 6.14231053055773284935287291528, 7.37616857535695739731354432730, 7.69303512813081196216099799400, 8.877311926556311718157585216342
