| L(s) = 1 | + 5·5-s + 22.5·7-s − 6.39·11-s + 78.5·13-s − 56.1·17-s + 82.9·19-s + 10.2·23-s + 25·25-s + 135.·29-s + 44.2·31-s + 112.·35-s + 128.·37-s − 174.·41-s − 270.·43-s + 596.·47-s + 167.·49-s − 0.289·53-s − 31.9·55-s − 655.·59-s + 48.7·61-s + 392.·65-s + 358.·67-s + 97.3·71-s − 617.·73-s − 144.·77-s + 448.·79-s − 723.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.21·7-s − 0.175·11-s + 1.67·13-s − 0.801·17-s + 1.00·19-s + 0.0929·23-s + 0.200·25-s + 0.869·29-s + 0.256·31-s + 0.545·35-s + 0.571·37-s − 0.663·41-s − 0.959·43-s + 1.85·47-s + 0.487·49-s − 0.000750·53-s − 0.0783·55-s − 1.44·59-s + 0.102·61-s + 0.749·65-s + 0.654·67-s + 0.162·71-s − 0.989·73-s − 0.213·77-s + 0.639·79-s − 0.957·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.239389832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.239389832\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 22.5T + 343T^{2} \) |
| 11 | \( 1 + 6.39T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 10.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 44.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 270.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 596.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.289T + 1.48e5T^{2} \) |
| 59 | \( 1 + 655.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 358.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 97.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 617.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 448.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 723.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 112.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 396.T + 9.12e5T^{2} \) |
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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
−8.962390187661455787161902885899, −8.455176276624170402813505621057, −7.66989436275914303851749250651, −6.65226096481712868977592850336, −5.83775254906170227105516882193, −5.00067207131618732558532909073, −4.15528441469696365237606685948, −2.99808152661762629957099906458, −1.79244550709812595813468615436, −0.966778151300225403399521368991,
0.966778151300225403399521368991, 1.79244550709812595813468615436, 2.99808152661762629957099906458, 4.15528441469696365237606685948, 5.00067207131618732558532909073, 5.83775254906170227105516882193, 6.65226096481712868977592850336, 7.66989436275914303851749250651, 8.455176276624170402813505621057, 8.962390187661455787161902885899
